Research ArticleMultistage Warning Indicators of Concrete Dam underInfluences of Random Factors
Guang Yang12 and Meng Yang13
1State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering Hohai University Nanjing 210098 China2National Engineering Research Center of Water Resources Efficient Utilization and Engineering Safety Hohai UniversityNanjing 210098 China3College of Water-Conservancy and Hydropower Hohai University Nanjing 210098 China
Correspondence should be addressed to Meng Yang ymym 059126com
Received 25 November 2015 Revised 25 February 2016 Accepted 21 March 2016
Academic Editor Paolo Crippa
Copyright copy 2016 G Yang and M YangThis is an open access article distributed under theCreativeCommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Warning indicators are required for the real-timemonitoring of the service conditions of dams to ensure safe andnormal operationsWarnings are traditionally targeted at some ldquosingle point deformationrdquo by deformation measuring points of concrete dam andscientific warning theory on ldquooverall deformationrdquo measured is nonexistent Furthermore the influences of random factors arenot considered In this paper the overall deformation of the dam was seen as a deformation system of single interactionalobservation points with different contribution degrees The spatial deformation entropy which describes the overall deformationwas established and the fuzziness indicator that measures the influence of complex random factors onmonitoring values accordingto cloud theory was constructed On this basis multistage warning indicators of ldquospatial deformationrdquo that consider fuzziness andrandomness were determined Analysis showed that the change law of information entropy of the damrsquo overall deformation isidentical to the real change law of the dam thus it reflects the real deformation state of the dam Moreover the identified warningindicators improved the warning ability of concrete dams
1 Introduction
Deformation is one of the major monitored items in damsafety Concrete dams are exposed to influences of variousnondeterministic settings such as the load effect of waterlevel uplift pressure and wind waves caused by hydrologicand hydraulic uncertainties as well as geological andmaterialuncertainties such as shearing and compressive strengthThus a concrete dam is a complicated system of nonde-terministic settings that are affected by various complexrandom factors [1 2] Considering a damrsquos long-term serviceconducting timely and effective warning against emergenciesthrough real-time monitoring is key to its safe operation [3]
The monitoring of dam safety is an important researchsubject in advanced mechanics and mathematics theories In1950 Tonini categorized the factors influencing the displace-ment of dam into water pressure temperature and effective-ness for a given period [4] These factors were expressed inthe polynomials of specific functions before a statistic model
with regression analysis was establishedThen the determin-istic model andmixedmodel were consulted for deformationof concrete dam and introduced finite element to monitorand evaluate the safety of dams [5] Furthermore manyscholars brought new achievements in diagnosing dam safetyfrom many aspects In 2009 Gu and Wang established thecatastrophe model of time-dependent component on thebasis of catastrophe theory and proposed the method todetermine the threshold value of the structure displacementof the dam [6] On the basis of the POT model in extremevalue theory in 2012 Su et al estimated the warningindicators by setting the threshold value and combiningthe probability of dam deformation with transfinite datasequence as the subject of modeling analysis [7] Warningindicators are sure to have some fuzziness and randomnessbecause of the influences of various complex random factorsOn the basis of the fuzzy finite element Chen (2006) realizedthe nondeterministic optimal control on roller compactedconcrete dam [8] Although the above-mentioned theories
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 6581204 12 pageshttpdxdoiorg10115520166581204
2 Mathematical Problems in Engineering
and methods complement and improve traditional methodsin solving difficult problems in dam safety such approachesonly address the warning of ldquopointrdquo deformation and not thescientific ldquospatial and overallrdquo deformation Further studieson the deformation of concrete dams must consider theinfluences of complex random factors Thus the expressionof a damrsquos overall deformation should be constructed andscientific and accurate warning indicators that considerrandomness and fuzziness should be determined
This paper began with an analysis of an indicator offuzziness that affects the value of monitoring the dam bystudying the influences of fuzziness and randomness fromlong-term service of the dam based on cloud theory [9ndash11] Thereafter the relationship between a damrsquos overalldeformation and deformation of single observation pointswas analyzed through information entropy and synergeticsIn the analysis overall deformation refers to a systemwhereinsingle observation points with different contribution degrees(weights) influence one another A criterion indicator mea-suring the overall deformation conditions was constructedOn this basis multistage warning indicators for the overalldeformation of concrete dam considering fuzziness andrandomness were determined Therefore nondeterministicoptimal control was achieved [12] This paper concludes witha project case that verified the feasibility of the proposedtheory
2 Nondeterministic Optimal Control
Hydraulic engineering considers that some fuzziness andrandomness in dams are inevitable because of the influencesof various random factors such as the nondeterminacy ofmechanical parameters imposed load and boundary condi-tions Statistically the smaller probability of dam displace-ment indicates that the dam is in a more dangerous state120583 + 3120590 and 120583 minus 3120590 can be used as a warning indicator in oneconfidence coefficient of the dam if the displacement obeysthe normal distribution of mean value and variance and is inthe range of 120583plusmn3120590 in deterministic optimal control In realitygiven that dams are affected by random factors warningindicators will have a range of variations For example inFigure 1 the warning value of upstream displacement has amaximum control value and minimum control value
As the foundation of cloud theory the cloud model isprecisely both controlled and uncontrolled in microscopescale119880 is the time series ofmonitoring the damdeformationand 119862 is the qualitative judgment on dam safety If thequantitative value 119909 isin 119880 119906(119909) fall in [0 1] and follow theprobability distribution law
120583 119880 997888rarr [0 1] forall119909 isin 119880 119909 997888rarr 120583 (119909) (1)
The distribution of 119909 in 119880 is called cloud and (119909 119906) is thecloud droplet
In Figure 2 for observation point 119894 if the range whereinthe cloud droplet falls is given the upper bound and lower
bound of the cloud droplet in the cloud model 119910119906119894and 119910119897
119894can
be expressed as follows
119910119906
119894(119909119894 119864119909119894 119864119899119894 119867119890119894) = 119890minus(119909119894minus119864119909119894)
22(119864119899119894+3119867119890119894)
2
119910119897
119894(119909119894 119864119909119894 119864119899119894 119867119890119894) = 119890minus(119909119894minus119864119909119894)
22(119864119899119894minus3119867119890119894)
2
(2)
119909119894119895is the value 119895 of observation point 119894 and the fuzziness
Δ119894119895can be calculated by
Δ119894119895= 119910119906
119894119895minus 119910119897
119894119895 (3)
In this equation 119910119906119894119895is the upper limit value of 119909
119894119895in the
range and 119910119897119894119895is the lower limit value of 119909
119894119895in the range
Given the influence of random factors when 119909119894119895ge 0
119909119894119895changes in the range of [119909
119894119895minus 119909119894119895Δ119894119895 119909119894119895+ 119909119894119895Δ119894119895] when
119909119894119895lt 0 it is in the range of [119909
119894119895+ 119909119894119895Δ119894119895 119909119894119895minus 119909119894119895Δ119894119895]
When drawing up the warning indicator for downstreamdeformation the maximum and minimum control values ofthe indicator can be determined when the significance level is120572 thus indicating that the nondeterministic optimal controlhas been achieved
3 Methods of Characterizing Contributions ofSingle Observation Point
The overall deformation condition of the concrete dam isusually exposed to the influences of water pressure and tem-perature and is related to many factors including the physicaland mechanical properties of dam materials body structuregeology and hydrology and it could be referred to in Figure 3The principles of synergetics posit that a concrete dam isa synthesis of feature points with different contributions(weights) that influence one another The contribution of asingle observation point needs to be studied to construct areasonable expression of overall deformation
31 Construction of the Indicator Set of a Single ObservationPoint Weight Entropy [13ndash15] a basic concept in thermody-namics refers to a state function in a system The conceptof information entropy is a measurement of the systemrsquosdisorder and nondeterminacy [15] The measured value 119895 onobservation point 119894 is 119909
119894119895and its corresponding entropy is 119878
119894119895
According to entropy theory when an observation point is ina more dangerous state the system is in greater disorder andits entropy value is smaller Thus the following is obtained
119878119894119895= minus [120583
119894119895ln 120583119894119895+ (1 minus 120583
119894119895) ln (1 minus 120583
119894119895)] (4)
120583119894119895=
int
119909119894119895
minusinfin
119891 (120589) 119889120589 119909119894119895ge 0
int
+infin
119909119894119895
119891 (120589) 119889120589 119909119894119895lt 0
(5)
Formula (4) defines the information entropy of themeasurement value No matter what distribution 119909
119894119895 obeys
if the probability density function of the measured value is
Mathematical Problems in Engineering 3
War
ning
val
ue o
f dow
nstre
am
disp
lace
men
t
War
ning
val
ue o
f ups
tream
di
spla
cem
ent
Displacement
Probability density
Displacement
The m
axim
um co
ntro
l val
ue o
f war
ning
va
lue o
f dow
nstre
am d
ispla
cem
ent
The m
inim
um co
ntro
l val
ue o
f war
ning
va
lue o
f dow
nstre
am d
ispla
cem
ent
The m
axim
um co
ntro
l val
ue o
f war
ning
va
lue o
f ups
tream
disp
lace
men
t
The m
inim
um co
ntro
l val
ue o
f war
ning
va
lue o
f ups
tream
disp
lace
men
t
Probability density
Figure 1 Diagram of deterministic optimal control and nondeterministic optimal control
Degree of certainty
Upper limit of cloud dropletsLower limit of cloud droplets
Cloud dropletsEx
Figure 2 Diagram of the range where the cloud droplet falls
known the corresponding information entropy sequence canbe calculated Deformation of dam can be divided into threeparts water pressure component temperature componentand aging component Aging component comprehensivelyreflects the creep and plastic deformation of dam concreteand rock foundation and compression deformation of geo-logical structure of rock foundation At the same time italso includes the irreversible displacement caused by the damcrack and the autogenous volume deformation It changesdramatically in the early stage and gradually tends to be stablein the later stage The project selected in this paper is a damwhich has worked for many years and its aging componentstend to be stable and the deformation value is stable in annualperiod rule which obeys normal distribution
The indicator measuring how much information is con-tained in 119909
119894119895is the inverse entropy 119863
119894119895 119863119894119895= 1 minus 119878
119894119895
When one measured value reflects more information theentropy value 119878
119894119895will be smaller and its inverse entropy 119863
119894119895
will be greater Thus 119863119894119895can be used to measure how much
information is reflected by a single measured value
Suppose the weight distribution of all observation pointsis 120596119894| 119894 = 1 2 119899 where 119899 is the number of points
observed and 120596119894will meet the following requirement 120596
119894ge 0
and sum120596119894= 1 The entropy 119878
119894119895of the object matrix of the
deformation measured value 119909119894119895| 119894 = 1 sim 119899 119895 = 1 sim 119898
can be computed through (4) and (5) The inverse entropymatrix is expressed as follows
119863119894119895=
[[[[[[
[
1198631111986312sdot sdot sdot 119863
1119898
1198632111986322sdot sdot sdot 119863
2119898
11986311989911198631198992sdot sdot sdot 119863
119899119898
]]]]]]
]
(6)
In this matrix 119863119894119895is the inverse entropy of 119909
119894119895 The weight
of the feature points was traced by the projection pursuitmethod
32 Process of Calculating the Weight of a Single Obser-vation Point By using the projection pursuit method [1617] the high-dimensional data can be projected to lowdimension space and projection that reflects the structureor features of high-dimensional data is pursued to analyzehigh-dimensional dataThismethod is advantageous becauseit is highly objective robust resistant to interference andaccurate The steps are as follows
Step 1 The extreme value of the inverse entropy matrix wasnormalized through the following equation
119863lowast
119894119895=
119863119894119895minus [119863119895]min
[119863119895]max minus [119863119895]min
(7)
where [119863119895]max and [119863119895]min are the maximum and minimum
values of line 119895 in the matrix respectively
4 Mathematical Problems in Engineering
Weight of surveypoints 1
Overall deformation of concrete dam
Deformation ofsurvey points 1
Deformation ofsurvey points n
Deformation ofsurvey points 2
Weight of surveypoints 2
Weight of surveypoints n
middot middot middot
middot middot middot
Figure 3 Diagram of overall deformation system of concrete dam
Step 2 The normalized value 119863lowast119894119895
was projected to unitdirection119875 119875 = (119901
1 1199012 119901
119895) and1199012
1+1199012
2+sdot sdot sdot+119901
2
119895= 1The
indicator function of the projection 119866(119894) was constructed
119866 (119894) =
119898
sum
119895=1
119901119895119863lowast
119894119895 (119894 = 1 2 119899) (8)
Step 3 The objective function of the projection was con-structed The best direction for the projection was estimatedby solving the maximization problem of the objective func-tion in the constraint condition
Objective function Max 119867 (119901) = 119878119866sdot 119876119866
Constraint condition119898
sum
119895=1
1199012
119895= 1
(9)
In this equation 119878119866is the divergent degree of the projection
119876119866is the local density of 1D data points along 119875 and is
expressed as follows
119878119866= [sum119899
119894=1(119866 (119894) minus 119892 (119894))
2
119899 minus 1]
05
119876119866=
119899
sum
119894=1
119899
sum
119895=1
(119877 minus 119903119894119895) sdot 119891 (119877 minus 119903
119894119895)
(10)
where119892(119894) is themean value of this sequence119877 is the windowradius of the local density and 119877 = 01119878
119866in this paper 119903
119894119895is
the distance between two projection values and 119891(119905) is theunit step function 119891(119905) is equal to 1 as 119905 is greater than 0Otherwise 119891(119905) is equal to 0
Step 4 The projection value of one sample point was com-puted by substituting the best direction 119875lowast into (8) 120596
119894can be
calculated by normalizing the projection value
120596119894=
119866lowast
(119894)
sum119899
119895=1119866lowast (119895)
119894 = 1 2 119899 (11)
4 Study on Equivalent Model ofDamrsquos Overall Deformation
The overall deformation of a dam can be considered a defor-mation system of feature points with different contributionsthat influence one another as well as observation points ofthe deformation as feature pointsThe deformation conditionwas analyzed systemically and the overall deformation wasexpressed by the evolution of equation of all feature pointsThe deformation condition was described qualitatively bythe tectonic type of information entropy The absolute valueof the information entropy measures the danger level ofthe deformation value Smaller absolute value means higherdanger level The positive and negative values indicate thedirection of the deformation A positive value correspondsto downstream deformation whereas a negative value corre-sponds to upstream deformation The influences of randomfactors were considered and the fuzzy information entropywas constructed
41 Constructing Fuzzy Information Entropy of SingleMeasured Value The downstream deformation is positivewhereas the upstream deformation is negative
When the observation point moves downstream make120583119894119895= int119909119894119895
minusinfin
119891(120589)119889120589 and according to the definition of infor-mation entropy the information entropy of 119909
119894119895is expressed
as (4)Considering the influence of Δ
119894119895 120583119894119895will float in [1205830
119894119895 1205831
119894119895]
the following is then obtained
1205830
119894119895= int
119909119894119895minus119909119894119895Δ 119894119895
minusinfin
119891 (120589) 119889120589
1205831
119894119895= int
119909119894119895+119909119894119895Δ 119894119895
minusinfin
119891 (120589) 119889120589
(12)
When the observation point moves upstream make 120583119894119895=
int+infin
119909119894119895
119891(120589)119889120589 the information entropy of 119909119894119895is defined as
follows119878119894119895= 120583119894119895ln 120583119894119895+ (1 minus 120583
119894119895) ln (1 minus 120583
119894119895) (13)
Mathematical Problems in Engineering 5
0
Dow
nstre
am d
ispla
cem
ent
Ups
tream
di
spla
cem
ent
105
Sij
1205830
1 minus 1205830 120583
Figure 4 Diagram of 119878119894119895changes with the change of 120583
119894119895if 119864119909ge 0
Considering the influence ofΔ119894119895 120583119894119895floats in [1205830
119894119895 1205831
119894119895]The
following is then obtained
1205830
119894119895= int
+infin
119909119894119895minus119909119894119895Δ 119894119895
119891 (120589) 119889120589
1205831
119894119895= int
+infin
119909119894119895+119909119894119895Δ 119894119895
119891 (120589) 119889120589
(14)
Considering the influences of random factors the infor-mation entropy of 119909
119894119895minus119878119894119895floats in [1198780
119894119895 1198781
119894119895] whichwas defined
as the fuzzy information entropy of 119909119894119895
Take the downstream as an example Influenced by deter-mined and random factors the fuzzy entropy of 119878
119894119895changes
into the range of 119878119894119895in the range of [1205830
119894119895 1205831
119894119895]
42 Methods for Determining the Range of the InformationEntropy of Single Measured Value For Figure 4 the expecta-tion of one deformation monitoring sequence sample at oneobservation point (119864
119909ge 0 119878
119894119895) changes with the change of
120583119894119895 as shown in Figure 4 where 120583
0= int0
minusinfin
119891(120589)119889120589When the dam deforms downstream the change law of
119878119894119895is as follows 119878
119894119895will increase with increasing 120583
119894119895when 120583
119894119895
is in the range of (1205830 05) 119878
119894119895will decrease with decreasing
120583119894119895when 120583
119894119895is in the range of (05 1) when 120583
119894119895= 05 119878
119894119895will
reach themaximumvalue If 05 is in the range of [1205830119894119895 1205831
119894119895] the
maximum of 119878119894119895is 1198781119894119895when 120583
119894119895= 05 and its minimum value
is 1198780119894119895at the endpoint If 05 is not in the range of [1205830
119894119895 1205831
119894119895]
119878119894119895will have its maximum value 1198781
119894119895and minimum value 1198780
119894119895
at endpoints When the dam deforms upstream 119878119894119895will rise
with the rise of 120583119894119895and it will have its maximum value 1198781
119894119895and
minimum value 1198780119894119895at endpoints
The expectation of one deformationmonitoring sequencesample at one observation point (119864
119909lt 0 119878
119894119895) changes with
01
05
Dow
nstre
amU
pstre
am d
ispla
cem
ent
Sij
1205830
1 minus 1205830 120583
disp
lace
men
t
Figure 5 Diagram of 119878119894119895changes with the change of 120583
119894119895if 119864119909lt 0
the change of 120583119894119895as shown in Figure 5 where 120583
0= int0
minusinfin
119891(120589)119889120589
and Figure 5 is presented for 119878119894119895changes with the change of
120583119894119895When the damdeforms downstream the change law of 119878
119894119895
is as follows 119878119894119895will increase with decreasing 120583
119894119895and will have
its maximum value 1198781119894119895and minimum value 1198780
119894119895at endpoints
When the damdeforms upstream the figure shows that when120583119894119895is in the range of (1minus120583
0 05) 119878
119894119895will increase with the drop
of 120583119894119895 when 120583
119894119895is in the range of (05 1) 119878
119894119895will decrease with
the increase of 120583119894119895 when 120583
119894119895= 05 119878
119894119895will reach theminimum
value If 05 is contained in the range of [1205830119894119895 1205831
119894119895] 119878119894119895will have
its minimum value 1198780119894119895at 120583119894119895= 05 and have its maximum
value 1198781119894119895at endpoint if 05 is not contained in the range 119878
119894119895
will have its maximum value 1198781119894119895and minimum value 1198780
119894119895at
endpoints
43 Construction of the Fuzzy Information Entropy of OverallDeformation On the basis of the above results the expres-sion of information entropy of overall deformation can bededucedThe contribution of the order degree of observationpoint 119894 is 120596
119894120583119894119895 make 1205831
119894119895= 120583119894119895and 1205832
119894119895= 1 minus 120583
119894119895 According to
the broad definition of information entropy when the dammove deforms downstream the expression of informationentropy of overall deformation is expressed as follows
119878119895= minus
119898
sum
119894=1
2
sum
119896=1
120596119894120583119896
119894119895ln (120596119894120583119896
119894119895)
= minus
119899
sum
119894=1
2
sum
119896=1
120596119894120583119896
119894119895(ln120596119894+ ln 120583119896
119894119895)
= minus
119899
sum
119894=1
2
sum
119896=1
120596119894120583119896
119894119895ln120596119894minus
119899
sum
119894=1
2
sum
119896=1
120596119894120583119896
119894119895ln 120583119896119894119895
6 Mathematical Problems in Engineering
Choice of sample
Input sample eigenvalues
Calling cloud model program
Generating range of cloud fall
Calculating the fuzzinessof all measured value
Constructing deformed entropyof one measured value
Calculating weight of singleobservation point
Constructing fuzzy deformedentropy of overall deformation
Constructing fuzzy deformedentropy of one measured value
Ex EnHe
Figure 6 Computational process of fuzzy information entropy of overall deformation
= minus
119899
sum
119894=1
120596119894ln120596119894
2
sum
119896=1
120583119896
119894119895minus
119899
sum
119894=1
120596119894
2
sum
119896=1
120583119896
119894119895ln 120583119896119894119895
= minus
119899
sum
119894=1
120596119894ln120596119894+
119899
sum
119894=1
120596119894119878119894119895
(15)
When the dam move deforms upstream the expressionof the information entropy of overall deformation is
119878119895=
119898
sum
119894=1
2
sum
119896=1
120596119894120583119896
119894119895ln (120596119894120583119896
119894119895) =
119899
sum
119894=1
120596119894ln120596119894+
119899
sum
119894=1
120596119894119878119894119895 (16)
Therefore the expression of information entropy of over-all deformation is defined as follows
119878119895=
minus
119899
sum
119894=1
120596119894ln120596119894+
119899
sum
119894=1
120596119894119878119894119895 119878119894119895ge 0
119899
sum
119894=1
120596119894ln120596119894+
119899
sum
119894=1
120596119894119878119894119895 119878
119894119895lt 0
(17)
The absolute value of the information entropy of the over-all deformationmeasures the danger level of the deformationthat is a smaller absolute value means a higher danger levelpositive and negative values stand for the direction of thedeformation a positive value means downstream deforma-tion whereas a negative value means upstream deformation
Considering the influences of random factors the fuzzyinformation entropy of 119909
119894119895is [1198780119894119895 1198781
119894119895] the fuzzy information
entropy of overall deformation can be illustrated through(17) Computational process of fuzzy information entropy ofoverall deformation is shown in Figure 6
5 Proposed Multistage FuzzyInformation Entropy of OverallDeformation Warning Indicators
Horizontal displacement of dam crest changes in an annualcycle ldquoupstream and downstream switchrdquo Therefore thisdisplacement should be in a certain scale and be controlledunder somemonitoring indicators for the safe damoperation
In the case of downstream displacement the primaryfuzzy warning indicator 1205751015840
1is defined as 1205751015840
1= (1205750
1 1205751
1) 12057501is
the lower limit of this indicator and 12057511is the upper limit
the secondary indicator 12057510158402is 12057510158402= (1205750
2 1205751
2) where 1205750
2is
the lower limit of this indicator and 12057512is the upper limit
When 12057511gt 1205750
2 a cross phenomenon appears in the primary
indicator and secondary indicator when both of them shouldbe categorized according to the membership of displacementmeasured 120575lowast was introduced because the membership ofdisplacement at this point is the same Figure 7 showsdiagram of multistage fuzzy information entropy warningindicators
The primary fuzzy warning indicator is 12057510158401= (1205750
1 120575lowast
) andthe secondary is 1205751015840
2= (120575lowast
1205751
2)
If the deformation value is in (12057501 1205751
1) or (1205750
1 120575lowast
) the damis in the state of primary warning if the value is in (1205750
2 1205751
2) or
(1205750
2 120575lowast
) the dam is in the state of secondary warningThe time sequence of deformation at each observation
point was analyzed by using the above theoretical methodThe lower and upper limits of the fuzzy information entropyof overall deformation affected by Δ
119894119895will be 1198780
119895 and 1198781
119895
Considering the damrsquos long-term service when the dammoves downstream the lower limit 1198780
119898119895 and upper limit
Mathematical Problems in Engineering 7
1Degree of membership
Displacement
1Degree of membership
Displacement
12057512120575021205751112057501
1205751212057502 1205751112057501 120575lowast
Figure 7Diagramofmultistage fuzzy information entropywarningindicators
1198781
119898119895 were selected and when the dam moves upstream
the lower limit 1198770119898119895 and upper limit 1198771
119898119895 were selected
1198780
119898119895 1198781119898119895 1198770119898119895 and 1198771
119898119895 are random variables and four
subsample spaces with the sample size of 119873 can be obtainedby the following
1198780
= 1198780
1198981 1198780
1198982 119878
0
119898119898
1198781
= 1198781
1198981 1198781
1198982 119878
1
119898119898
1198770
= 1198770
1198981 1198770
1198982 119877
0
119898119898
1198771
= 1198771
1198981 1198771
1198982 119877
1
119898119898
(18)
Shapiro-Wilk test andKolmogorov-Smirnov test can bothtest whether the samples obey normal distribution or not Butthe Kolmogorov-Smirnov test is applicable to fewer samplesIt can not only test if the samples are subject to normaldistribution but also test if samples are subject to otherdistributions The basic idea of the K-S test is to comparethe cumulative frequency of the observed value (119865
119899(119909)) with
the assumed theoretical probability distribution (119865119909(119909)) to
construct statisticsAccording to the method of empirical distribution func-
tion segmented cumulative frequency is obtained by usingthe following formula
119865119899(119909) =
0 119909 lt 119909119894
119894
119899 119909119894le 119909 lt 119909
119894+1
1 119909 ge 119909
(19)
In the formula 1199091 1199092 119909
119899is sample data after arrange-
ment The sample size is 119899
In the full range of random variable 119883 the maximumdifference between 119865
119899(119909) and 119865
119909(119909) is
119863119899= max 1003816100381610038161003816119865119909 (119909) minus 119865119899 (119909)
1003816100381610038161003816 lt 119863120573
119899 (20)
In the formula 119863119899is a random variable whose distribu-
tion depends on 119899119863120573119899is critical value for a significant level 120573
It is considered that the distribution to be used at a significantlevel 120573 cannot be resisted otherwise it should be rejected
The distribution formwas tested through the K-Smethodto determine the probability density function Fuzzy warningindicator was then determinedwith different significant levelIn dam safety evaluation significant level 120572 is the probabilityof the dam failure Supposing 119878
119898is the extreme of the
information entropy of the upstream overall deformation if119878 gt 119878
119898 the probability of the dam failure is 119875(119878 gt 119878
119898) =
120572 = intinfin
119878119898
119891(119909)119889119909 and the reliability index of dam failure is1 minus 120572 According to the dam importance different failureprobability is set and the multistage warning indicators wereidentified
6 Example Analysis
61 Project Profile One flat-slab deck dam built with rein-forced concrete is an important part of one river basin cascadeexploitation The elevation of this dam crest is 13770m andthe height of biggest part is about 43m the crest runs 2250min length and is made of 27 flat-slab buttresses with a span of75m The space between the left side of 2 buttress and theright side of 9 buttress is the joint part the overflow buttressis located from the 9 buttress to the 20 buttress the rest isthe water-retaining buttressTheworkshop buttress is locatedfrom the 5 buttress to the 8 buttress In this dam the levelof deadwater is 1220m the normal highwater level is 1310m(in practice it is 1290m) the design flood level is 1367m andthe check flood level is 1375m To monitor the displacementof this dam a direct plumb line and an inverted plumb linewere arranged in four buttresses 4 9 21 and 24There isa crushed zone under the dam foundation where occurrenceis N20∘sim25∘W SWang70∘sim80∘ the maximum width is about3m and the narrowest place is about 1mThere is an elevationclip joint mud at level 91m
The deformation field characterized by the observationpoint at 21 should be typical and is a key point because it isin the riverbedThus the observation point G21 at the heightof 134m along the direct plumb line at 21 was analyzed aswell as point 27 at the height of 118m along this line andpoint 28 at the height of 107mAll these valuesmeasuredweretransformed into absolute displacement The arrangement ofeach point is shown in Figure 8 The daily monitoring dataseries is from January 1 2003 to December 31 2013
In this paper two-stage warning indicators were setaccording to the practical running of this project and danger120572 = 5 is the primary warning that is mainly used to discri-minate and handle early dangerous case and the reliabilityindex of dam failure is 95 whereas 120572 = 1 is the secondarywarning that is mainly used to determine grave danger andprevent urgent danger and the reliability index of dam failureis 99
8 Mathematical Problems in Engineering
G21 G9 G4G24
25
30
2631
28
27
3233
Figure 8 The arrangement of observation point
Ups
tream
wat
er le
vel
623
198
4
121
419
89
66
1995
112
620
00
519
200
6
119
201
1
11
1979
Date
120124128132
Figure 9 The process line of upstream water level
11
2007
11
2008
11
2006
123
120
08
11
2005
123
120
09
123
120
10
123
120
11
123
020
12
123
020
13
Data
05
101520253035
Tem
pera
ture
Figure 10 The process line of temperature
62 Calculating the Contribution of Deformation at theObservation Point to the Overall Deformation Figures 9-10 show the upstream stage hydrograph and temperaturestage hydrograph respectively The water stage remainedunchanged whereas the temperature changed in the annualcycle Figure 11 shows the relationship between informationentropy of overall deformation and temperature in 2007 Anegative correlation exists between the overall deformationof 21 buttress and temperature that is an increase in tem-perature corresponds to the decrease in upstream or down-stream displacement and a decrease in temperature meansan increase in the upstream or downstream displacementFigure 12 shows the correlation between the displacementat observation point G21 and the temperature The overalldeformation law is roughly identical with that at observationpoint G21
Figure 12 reveals that the temperature obviously influ-enced overall deformation that is the temperature canchange the contribution of single observation point to theoverall deformation Temperature change was divided intothe stage of temperature rise and stage of temperature drop
2011
11
2011
21
2011
31
2011
41
2011
51
2011
61
2011
71
2011
81
2011
91
2011
10
1
2011
11
1
2011
12
1
TemperatureDeformed entropy
0070
140210280
Tem
pera
ture
minus080minus076minus072minus068minus064minus060
Def
orm
ed en
tropy
Figure 11 The relationship between information entropy of overalldeformation and temperature in 2011
2011
11
2011
13
1
2011
32
2011
41
2011
51
2011
53
1
2011
63
0
2011
73
0
2011
82
9
2011
92
8
2011
10
28
2011
11
27
2011
12
27
Horizontal displacementTemperature
05
101520253035
Tem
pera
ture
minus16minus12minus08minus04004
Hor
izon
tal
disp
lace
men
t
Figure 12 The relationship between horizontal displacement andtemperature of observation point G21 in 2011
The weight of deformation at observation point in two stageswas calculated The results are shown in the Table 1
63 Results of Information Entropy of Overall DeformationRange of cloud fall of the displacement at G21 27 and28 is shown in Figure 13 The boundary values of mostdangerous information entropy from 2003 to 2013 are shownin Tables 2ndash5The absolute value means the degree of dangerdownstream is set as negative value In the significance level120572 = 005 7 kinds of common distribution (lognormal distri-bution normal distribution uniform distribution triangulardistribution exponential distribution 120574 distribution and 120573distribution) were used to carry on hypothesis test for 1198780
119898119895
1198781
119898119895 1198770119898119895 and 1198771
119898119895 with K-S method The maximum 119863
119899
of each sequencewas obtained and comparedwith the criticalvalue 119863005
119899of the significant level 005 to determine the type
Mathematical Problems in Engineering 9
Table 1 The weight table of observation point
Observation point Altitude The period of temperature rise The period of temperature dropG21 134m 0357 039227 118m 0331 031628 107m 0312 0292
Table 2 The lower limit of the most dangerous information entropy of downstream overall deformation
Year 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013Information entropy 04784 04837 04767 04784 04969 04887 04969 05102 05082 04799 04739
Table 3 The upper limit of the most dangerous information entropy of downstream overall deformation
Year 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013Information entropy 04791 04845 04770 04786 04971 04899 04974 05186 05161 04817 04769
Table 4 The lower limit of the most dangerous information entropy of upstream overall deformation
Year 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013Information entropy minus05433 minus05491 minus04773 minus05083 minus05067 minus04784 minus05080 minus04839 minus04836 minus04825 minus05020
Table 5 The lower limit of the most dangerous information entropy of upstream overall deformation
Year 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013Information entropy minus05339 minus05440 minus04770 minus05079 minus05055 minus04770 minus05040 minus04822 minus04806 minus04754 minus04766
Cloud droplets
minus15
minus14
minus13
minus12
minus11
minus1
minus09
minus08
minus07
minus06
minus05
minus04
minus03
minus02
minus01
010
02
03
04
05
06
07
08
09 1
00102030405060708091
Cer
tain
ty
G21 27 28
Figure 13 Range of cloud fall of the displacement at G21 27 and 28
of the best distribution K-S test results are shown in Table 6Multistage fuzzy warning values are presented in Table 7
K-S test shows that 1198780119898119895 1198781119898119895 1198770119898119895 and 1198771
119898119895 satisfy
normal distributionThe probability density function of the sequence is
119891 (119909) =1
radic21205871205902exp(minus
(119909 minus 120583)2
120590) (21)
Parameter values of (21) are presented in Table 7In downstream deformation if 120572 = 5 the primary
warning indicator is (04763 04775) if 120572 = 1 thesecondary warning indicator is (04757 04763) 120575lowast is used as
the boundary value when two indicators overlap In the caseof upstream deformation if 120572 = 5 the primary warningindicator is (minus04779 minus04768) if 120572 = 1 the secondarywarning indicator is (minus04768 minus04763) (Table 8)
64 The Structure Calculation of Monitoring Index of DamHorizontal Displacement According to the actual situationthree-dimensional finite element model of the dam is estab-lished According to the structure and basic geologicalconditions of 21 dam section the scope of finite elementcalculation model can be got as follows taking 15 times ashigh dam in the upstream direction taking 15 times as highdam in the upstream direction and taking 1 time as highdam below the dam foundation The model is constitutedof 11445 nodes and 8571 units The unit type is 6 sides 8nodes isoparametric element The deformation observationdata analysis shows the dam under the condition of lowtemperature and highwater level there is larger displacementin the downstream when in high temperature and in lowwater level there is larger displacement in the upstream(Table 9 Figure 14) In view of the actual situation of thedam observation and data analysis results the load conditionselection is as follows (Table 10 Figures 15 and 16)
Working condition 1 normal water level 1290m andmaximum temperature dropWorking condition 2 dead water level 1220m andmaximum temperature rise
10 Mathematical Problems in Engineering
Table 6 K-S test results
Probability distribution 1198780
1198981198951198781
1198981198951198770
1198981198951198771
119898119895
Lognormal distribution 026 011 023 028Normal distribution 011 008 026 022Uniform distribution 074 155 088 068Triangular distribution 053 054 059 061Exponential distribution 041 042 055 035120574 distribution 037 036 031 033120573 distribution 068 073 086 063119863005
119899034 034 029 029
The most reasonable probability distribution Normal distribution Normal distribution Normal distribution Normal distribution
Table 7 Parameter values of the probability density function
Data series Parameter values120583 120590
2
1198780
119898119895 0488355 00129
1198781
119898119895 0490627 00151
1198770
119898119895 minus050210 00249
1198771
119898119895 minus049674 00245
Figure 14 Finite element model of the dam
The primary warning indicators of concrete dam wereobtained by calculation methods for structures If the infor-mation entropy of the overall deformation reached 04770the dammoveddownstreamand in the state of primarywarn-ing If the information entropy of the overall deformationreached minus04773 the dam moved upstream and in the stateof primary warning (Table 11) They all fell into intervalscalculated by fuzzy methodologyThe analysis shows that themethod brought up in this paper is reasonable and scientificAlso the analysis shows the physical meaning of the fuzzywarning indexUnder the action of the unfavorable load com-bination and the influence of the complex random factorsthe maximum entropy andminimum information entropy ofthe overall deformation lie in this interval Considering theinfluences of random factorsmultistage fuzzywarning valueswere receptive and safe
In this paper the overall deformation of 21 buttresswas analyzed through the theoretical method proposedInfluences of random factors on the warning value wereconsidered and multistage fuzzy warning values were deter-mined If the information entropy of overall deformationis in (04763 04775) the dam moved downstream and in
minus2399e minus 003minus1296e minus 003minus1940e minus 0049084e minus 0042011e minus 0033113e minus 0034216e minus 0035318e minus 0036421e minus 0037523e minus 0038625e minus 003
XY
Z
Figure 15The results of finite element calculation ofworking condi-tion 1
XY
Z minus6007e minus 004minus5014e minus 004minus4021e minus 004minus3028e minus 004minus2035e minus 004minus1042e minus 004minus4872e minus 0069443e minus 0051937e minus 0042930e minus 0043923e minus 004
Figure 16 The results of finite element calculation of workingcondition 2
the state of primary warning If the information entropy ofoverall deformation is in (04757 04763) the dam moveddownstream and in the state of secondary warning If theinformation entropy of overall deformation is in (minus04779minus04768) the dam moved upstream and in the state ofprimary warning If the information entropy of the over-all deformation is in (minus04768 minus04763) the dam movedupstream and in the state of secondary warning
7 Conclusion
This paper presented multistage warning indicators of con-crete dam space and considered the influences of complex
Mathematical Problems in Engineering 11
Table 8 Multistage fuzzy warning values of concrete dam
The direction of deformation Confidence levelThe primary warning indicator The secondary warning indicator
Downstream (04763 04775) (04757 04763)Upstream (minus04779 minus04768) (minus04768 minus04763)
Table 9 Material parameter of the dam
Structure Density (kgm) Poissonrsquos ratio Elastic modulus (GPa)Concrete face slab 2400 0167 24Buttress 2400 0167 24Partition wall 2400 0167 24Reinforced concrete block 2400 0160 22Foundation rock mass 2700 0175 12Diorite-dyke 2000 03 115Crushed zone 2000 03 029Horizontal joints 2000 03 07
Table 10 The results of finite element calculation of displacementof observation point (mm)
Observation point Working condition 1 Working condition 2G21 0785 minus059927 0654 minus038628 0356 0114
Table 11 The primary warning indicators of concrete dam
The direction of deformation The primary warning indicatorDownstream 04770Upstream minus04773
random factors The results of the specific studies are asfollows
(1) Influences of fuzziness and randomness of randomfactors on the long-term service of dam were dis-cussed a fuzziness indicator that measures the influ-ence of random factors on monitoring value wasconstructed through cloud model
(2) Equivalent model of overall deformation was pro-posed In the model the overall deformation ofdam was regarded as a deformation system whereeach observation point had different contributions(weights) and affected one another Based on entropytheory a space information entropy that can measurethe overall deformation condition was established
(3) Multistage warning indicators against overall defor-mation of concrete dam under the influences of fuzzi-ness and randomness were determined and nondeter-ministic optimal control of the indicator was achievedto improve the competence of warning against thedeformation of concrete dam
Competing Interests
No conflict of interests exits in the submission of this paper
Acknowledgments
This paper was financially supported by National Natu-ral Science Foundation of China (Grants nos 4132300151139001 51579086 51379068 51279052 51579083 and51209077) Jiangsu Natural Science Foundation (Grants nosBK20140039 and BK2012036) Research Fund for the Doc-toral Program of Higher Education of China (Grant no20130094110010) the Ministry of Water Resources PublicWelfare Industry Research Special Fund Project (Grantsnos 201201038 and 201301061) Jiangsu Province ldquo333 High-Level Personnel Training Projectrdquo (Grants nos BRA2011179and BRA2011145) Project Funded by the Priority AcademicProgram Development of Jiangsu Higher Education Institu-tions (Grant no YS11001) Jiangsu Province ldquo333 High-LevelPersonnel Training Projectrdquo (Grant no 2017-B08037) JiangsuProvince ldquoSix Talent Peaksrdquo Project (Grant no JY-008) andFundamental Research Funds for the Central Universities(Grants nos 2016B04114 and 2015B25414)
References
[1] F BenedettoGGiunta andLMastroeni ldquoAmaximumentropymethod to assess the predictability of financial and commoditypricesrdquo Digital Signal Processing vol 46 pp 19ndash31 2015
[2] S Muraro A Madaschi and A Gajo ldquoOn the reliability of 3Dnumerical analyses on passive piles used for slope stabilisationin frictional soilsrdquo Geotechnique vol 64 no 6 pp 486ndash4922014
[3] W J Yoon and K S Park ldquoA study on the market instabilityindex and risk warning levels in early warning system foreconomic crisisrdquo Digital Signal Processing vol 29 pp 35ndash442014
[4] D Tonini ldquoObserved behavior of several leakier arch damsrdquoJournal of the Power Division vol 82 no 12 pp 135ndash139 1956
12 Mathematical Problems in Engineering
[5] R Salgado and D Kim ldquoReliability analysis of load andresistance factor design of slopesrdquo Journal of Geotechnical andGeoenvironmental Engineering vol 140 no 1 pp 57ndash73 2014
[6] Y C Gu and S J Wang ldquoStudy on the early-warning thresholdof structural instability unexpected accidents of reservoir damrdquoJournal of Hydraulic Engineering vol 40 no 12 pp 1467ndash14722009
[7] H Z Su F Wang and H P Liu ldquoEarly-warning index for damservice behavior based on POT modelrdquo Journal of HydraulicEngineering vol 43 no 8 pp 974ndash986 2012
[8] L Chen Mechanics performance with parameters changing inspace amp monitoring model of roller compacted concrete dam[PhD thesis] Hohai University 2006
[9] A B Li L F Zhang Q L Wang B Li Z Li and Y WangldquoInformation theory in nonlinear error growth dynamics andits application to predictability taking the Lorenz system as anexamplerdquo Science China Earth Sciences vol 56 no 8 pp 1413ndash1421 2013
[10] O Victor ldquoThe theory of cloudsrdquo Library Journal vol 132 no13 p 62 2007
[11] M Yang H Z Su and X Q Yan ldquoComputation and analysis ofhigh rocky slope safety in a water conservancy projectrdquoDiscreteDynamics in Nature and Society vol 2015 Article ID 197579 11pages 2015
[12] C Ricotta and G C Avena ldquoEvaluating the degree of fuzzi-ness of thematic maps with a generalized entropy functiona methodological outlookrdquo International Journal of RemoteSensing vol 23 no 20 pp 4519ndash4523 2002
[13] V I Khvorostyanov and J A Curry ldquoParameterization ofhomogeneous ice nucleation for cloud and climate modelsbased on classical nucleation theoryrdquo Atmospheric Chemistryand Physics vol 12 no 19 pp 9275ndash9302 2012
[14] H Z Su Z P Wen and Z R Wu ldquoStudy on an intelligentinference engine in early-warning system of damhealthrdquoWaterResources Management vol 25 no 6 pp 1545ndash1563 2011
[15] M Kaminski ldquoProbabilistic entropy in homogenization ofthe periodic fiber-reinforced composites with random elasticparametersrdquo International Journal for Numerical Methods inEngineering vol 90 no 8 pp 939ndash954 2012
[16] E Czogała and J Łeski ldquoApplication of entropy and energymeasures of fuzziness to processing of ECG signalrdquo Fuzzy Setsand Systems vol 97 no 1 pp 9ndash18 1998
[17] K Wu and J L Jin ldquoProjection pursuit model for evaluation ofregion water resource security based on changeable weight andinformationrdquo Resources and Environment in the Yangtze Basinvol 20 no 9 pp 1085ndash1091 2011
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
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Stochastic AnalysisInternational Journal of
2 Mathematical Problems in Engineering
and methods complement and improve traditional methodsin solving difficult problems in dam safety such approachesonly address the warning of ldquopointrdquo deformation and not thescientific ldquospatial and overallrdquo deformation Further studieson the deformation of concrete dams must consider theinfluences of complex random factors Thus the expressionof a damrsquos overall deformation should be constructed andscientific and accurate warning indicators that considerrandomness and fuzziness should be determined
This paper began with an analysis of an indicator offuzziness that affects the value of monitoring the dam bystudying the influences of fuzziness and randomness fromlong-term service of the dam based on cloud theory [9ndash11] Thereafter the relationship between a damrsquos overalldeformation and deformation of single observation pointswas analyzed through information entropy and synergeticsIn the analysis overall deformation refers to a systemwhereinsingle observation points with different contribution degrees(weights) influence one another A criterion indicator mea-suring the overall deformation conditions was constructedOn this basis multistage warning indicators for the overalldeformation of concrete dam considering fuzziness andrandomness were determined Therefore nondeterministicoptimal control was achieved [12] This paper concludes witha project case that verified the feasibility of the proposedtheory
2 Nondeterministic Optimal Control
Hydraulic engineering considers that some fuzziness andrandomness in dams are inevitable because of the influencesof various random factors such as the nondeterminacy ofmechanical parameters imposed load and boundary condi-tions Statistically the smaller probability of dam displace-ment indicates that the dam is in a more dangerous state120583 + 3120590 and 120583 minus 3120590 can be used as a warning indicator in oneconfidence coefficient of the dam if the displacement obeysthe normal distribution of mean value and variance and is inthe range of 120583plusmn3120590 in deterministic optimal control In realitygiven that dams are affected by random factors warningindicators will have a range of variations For example inFigure 1 the warning value of upstream displacement has amaximum control value and minimum control value
As the foundation of cloud theory the cloud model isprecisely both controlled and uncontrolled in microscopescale119880 is the time series ofmonitoring the damdeformationand 119862 is the qualitative judgment on dam safety If thequantitative value 119909 isin 119880 119906(119909) fall in [0 1] and follow theprobability distribution law
120583 119880 997888rarr [0 1] forall119909 isin 119880 119909 997888rarr 120583 (119909) (1)
The distribution of 119909 in 119880 is called cloud and (119909 119906) is thecloud droplet
In Figure 2 for observation point 119894 if the range whereinthe cloud droplet falls is given the upper bound and lower
bound of the cloud droplet in the cloud model 119910119906119894and 119910119897
119894can
be expressed as follows
119910119906
119894(119909119894 119864119909119894 119864119899119894 119867119890119894) = 119890minus(119909119894minus119864119909119894)
22(119864119899119894+3119867119890119894)
2
119910119897
119894(119909119894 119864119909119894 119864119899119894 119867119890119894) = 119890minus(119909119894minus119864119909119894)
22(119864119899119894minus3119867119890119894)
2
(2)
119909119894119895is the value 119895 of observation point 119894 and the fuzziness
Δ119894119895can be calculated by
Δ119894119895= 119910119906
119894119895minus 119910119897
119894119895 (3)
In this equation 119910119906119894119895is the upper limit value of 119909
119894119895in the
range and 119910119897119894119895is the lower limit value of 119909
119894119895in the range
Given the influence of random factors when 119909119894119895ge 0
119909119894119895changes in the range of [119909
119894119895minus 119909119894119895Δ119894119895 119909119894119895+ 119909119894119895Δ119894119895] when
119909119894119895lt 0 it is in the range of [119909
119894119895+ 119909119894119895Δ119894119895 119909119894119895minus 119909119894119895Δ119894119895]
When drawing up the warning indicator for downstreamdeformation the maximum and minimum control values ofthe indicator can be determined when the significance level is120572 thus indicating that the nondeterministic optimal controlhas been achieved
3 Methods of Characterizing Contributions ofSingle Observation Point
The overall deformation condition of the concrete dam isusually exposed to the influences of water pressure and tem-perature and is related to many factors including the physicaland mechanical properties of dam materials body structuregeology and hydrology and it could be referred to in Figure 3The principles of synergetics posit that a concrete dam isa synthesis of feature points with different contributions(weights) that influence one another The contribution of asingle observation point needs to be studied to construct areasonable expression of overall deformation
31 Construction of the Indicator Set of a Single ObservationPoint Weight Entropy [13ndash15] a basic concept in thermody-namics refers to a state function in a system The conceptof information entropy is a measurement of the systemrsquosdisorder and nondeterminacy [15] The measured value 119895 onobservation point 119894 is 119909
119894119895and its corresponding entropy is 119878
119894119895
According to entropy theory when an observation point is ina more dangerous state the system is in greater disorder andits entropy value is smaller Thus the following is obtained
119878119894119895= minus [120583
119894119895ln 120583119894119895+ (1 minus 120583
119894119895) ln (1 minus 120583
119894119895)] (4)
120583119894119895=
int
119909119894119895
minusinfin
119891 (120589) 119889120589 119909119894119895ge 0
int
+infin
119909119894119895
119891 (120589) 119889120589 119909119894119895lt 0
(5)
Formula (4) defines the information entropy of themeasurement value No matter what distribution 119909
119894119895 obeys
if the probability density function of the measured value is
Mathematical Problems in Engineering 3
War
ning
val
ue o
f dow
nstre
am
disp
lace
men
t
War
ning
val
ue o
f ups
tream
di
spla
cem
ent
Displacement
Probability density
Displacement
The m
axim
um co
ntro
l val
ue o
f war
ning
va
lue o
f dow
nstre
am d
ispla
cem
ent
The m
inim
um co
ntro
l val
ue o
f war
ning
va
lue o
f dow
nstre
am d
ispla
cem
ent
The m
axim
um co
ntro
l val
ue o
f war
ning
va
lue o
f ups
tream
disp
lace
men
t
The m
inim
um co
ntro
l val
ue o
f war
ning
va
lue o
f ups
tream
disp
lace
men
t
Probability density
Figure 1 Diagram of deterministic optimal control and nondeterministic optimal control
Degree of certainty
Upper limit of cloud dropletsLower limit of cloud droplets
Cloud dropletsEx
Figure 2 Diagram of the range where the cloud droplet falls
known the corresponding information entropy sequence canbe calculated Deformation of dam can be divided into threeparts water pressure component temperature componentand aging component Aging component comprehensivelyreflects the creep and plastic deformation of dam concreteand rock foundation and compression deformation of geo-logical structure of rock foundation At the same time italso includes the irreversible displacement caused by the damcrack and the autogenous volume deformation It changesdramatically in the early stage and gradually tends to be stablein the later stage The project selected in this paper is a damwhich has worked for many years and its aging componentstend to be stable and the deformation value is stable in annualperiod rule which obeys normal distribution
The indicator measuring how much information is con-tained in 119909
119894119895is the inverse entropy 119863
119894119895 119863119894119895= 1 minus 119878
119894119895
When one measured value reflects more information theentropy value 119878
119894119895will be smaller and its inverse entropy 119863
119894119895
will be greater Thus 119863119894119895can be used to measure how much
information is reflected by a single measured value
Suppose the weight distribution of all observation pointsis 120596119894| 119894 = 1 2 119899 where 119899 is the number of points
observed and 120596119894will meet the following requirement 120596
119894ge 0
and sum120596119894= 1 The entropy 119878
119894119895of the object matrix of the
deformation measured value 119909119894119895| 119894 = 1 sim 119899 119895 = 1 sim 119898
can be computed through (4) and (5) The inverse entropymatrix is expressed as follows
119863119894119895=
[[[[[[
[
1198631111986312sdot sdot sdot 119863
1119898
1198632111986322sdot sdot sdot 119863
2119898
11986311989911198631198992sdot sdot sdot 119863
119899119898
]]]]]]
]
(6)
In this matrix 119863119894119895is the inverse entropy of 119909
119894119895 The weight
of the feature points was traced by the projection pursuitmethod
32 Process of Calculating the Weight of a Single Obser-vation Point By using the projection pursuit method [1617] the high-dimensional data can be projected to lowdimension space and projection that reflects the structureor features of high-dimensional data is pursued to analyzehigh-dimensional dataThismethod is advantageous becauseit is highly objective robust resistant to interference andaccurate The steps are as follows
Step 1 The extreme value of the inverse entropy matrix wasnormalized through the following equation
119863lowast
119894119895=
119863119894119895minus [119863119895]min
[119863119895]max minus [119863119895]min
(7)
where [119863119895]max and [119863119895]min are the maximum and minimum
values of line 119895 in the matrix respectively
4 Mathematical Problems in Engineering
Weight of surveypoints 1
Overall deformation of concrete dam
Deformation ofsurvey points 1
Deformation ofsurvey points n
Deformation ofsurvey points 2
Weight of surveypoints 2
Weight of surveypoints n
middot middot middot
middot middot middot
Figure 3 Diagram of overall deformation system of concrete dam
Step 2 The normalized value 119863lowast119894119895
was projected to unitdirection119875 119875 = (119901
1 1199012 119901
119895) and1199012
1+1199012
2+sdot sdot sdot+119901
2
119895= 1The
indicator function of the projection 119866(119894) was constructed
119866 (119894) =
119898
sum
119895=1
119901119895119863lowast
119894119895 (119894 = 1 2 119899) (8)
Step 3 The objective function of the projection was con-structed The best direction for the projection was estimatedby solving the maximization problem of the objective func-tion in the constraint condition
Objective function Max 119867 (119901) = 119878119866sdot 119876119866
Constraint condition119898
sum
119895=1
1199012
119895= 1
(9)
In this equation 119878119866is the divergent degree of the projection
119876119866is the local density of 1D data points along 119875 and is
expressed as follows
119878119866= [sum119899
119894=1(119866 (119894) minus 119892 (119894))
2
119899 minus 1]
05
119876119866=
119899
sum
119894=1
119899
sum
119895=1
(119877 minus 119903119894119895) sdot 119891 (119877 minus 119903
119894119895)
(10)
where119892(119894) is themean value of this sequence119877 is the windowradius of the local density and 119877 = 01119878
119866in this paper 119903
119894119895is
the distance between two projection values and 119891(119905) is theunit step function 119891(119905) is equal to 1 as 119905 is greater than 0Otherwise 119891(119905) is equal to 0
Step 4 The projection value of one sample point was com-puted by substituting the best direction 119875lowast into (8) 120596
119894can be
calculated by normalizing the projection value
120596119894=
119866lowast
(119894)
sum119899
119895=1119866lowast (119895)
119894 = 1 2 119899 (11)
4 Study on Equivalent Model ofDamrsquos Overall Deformation
The overall deformation of a dam can be considered a defor-mation system of feature points with different contributionsthat influence one another as well as observation points ofthe deformation as feature pointsThe deformation conditionwas analyzed systemically and the overall deformation wasexpressed by the evolution of equation of all feature pointsThe deformation condition was described qualitatively bythe tectonic type of information entropy The absolute valueof the information entropy measures the danger level ofthe deformation value Smaller absolute value means higherdanger level The positive and negative values indicate thedirection of the deformation A positive value correspondsto downstream deformation whereas a negative value corre-sponds to upstream deformation The influences of randomfactors were considered and the fuzzy information entropywas constructed
41 Constructing Fuzzy Information Entropy of SingleMeasured Value The downstream deformation is positivewhereas the upstream deformation is negative
When the observation point moves downstream make120583119894119895= int119909119894119895
minusinfin
119891(120589)119889120589 and according to the definition of infor-mation entropy the information entropy of 119909
119894119895is expressed
as (4)Considering the influence of Δ
119894119895 120583119894119895will float in [1205830
119894119895 1205831
119894119895]
the following is then obtained
1205830
119894119895= int
119909119894119895minus119909119894119895Δ 119894119895
minusinfin
119891 (120589) 119889120589
1205831
119894119895= int
119909119894119895+119909119894119895Δ 119894119895
minusinfin
119891 (120589) 119889120589
(12)
When the observation point moves upstream make 120583119894119895=
int+infin
119909119894119895
119891(120589)119889120589 the information entropy of 119909119894119895is defined as
follows119878119894119895= 120583119894119895ln 120583119894119895+ (1 minus 120583
119894119895) ln (1 minus 120583
119894119895) (13)
Mathematical Problems in Engineering 5
0
Dow
nstre
am d
ispla
cem
ent
Ups
tream
di
spla
cem
ent
105
Sij
1205830
1 minus 1205830 120583
Figure 4 Diagram of 119878119894119895changes with the change of 120583
119894119895if 119864119909ge 0
Considering the influence ofΔ119894119895 120583119894119895floats in [1205830
119894119895 1205831
119894119895]The
following is then obtained
1205830
119894119895= int
+infin
119909119894119895minus119909119894119895Δ 119894119895
119891 (120589) 119889120589
1205831
119894119895= int
+infin
119909119894119895+119909119894119895Δ 119894119895
119891 (120589) 119889120589
(14)
Considering the influences of random factors the infor-mation entropy of 119909
119894119895minus119878119894119895floats in [1198780
119894119895 1198781
119894119895] whichwas defined
as the fuzzy information entropy of 119909119894119895
Take the downstream as an example Influenced by deter-mined and random factors the fuzzy entropy of 119878
119894119895changes
into the range of 119878119894119895in the range of [1205830
119894119895 1205831
119894119895]
42 Methods for Determining the Range of the InformationEntropy of Single Measured Value For Figure 4 the expecta-tion of one deformation monitoring sequence sample at oneobservation point (119864
119909ge 0 119878
119894119895) changes with the change of
120583119894119895 as shown in Figure 4 where 120583
0= int0
minusinfin
119891(120589)119889120589When the dam deforms downstream the change law of
119878119894119895is as follows 119878
119894119895will increase with increasing 120583
119894119895when 120583
119894119895
is in the range of (1205830 05) 119878
119894119895will decrease with decreasing
120583119894119895when 120583
119894119895is in the range of (05 1) when 120583
119894119895= 05 119878
119894119895will
reach themaximumvalue If 05 is in the range of [1205830119894119895 1205831
119894119895] the
maximum of 119878119894119895is 1198781119894119895when 120583
119894119895= 05 and its minimum value
is 1198780119894119895at the endpoint If 05 is not in the range of [1205830
119894119895 1205831
119894119895]
119878119894119895will have its maximum value 1198781
119894119895and minimum value 1198780
119894119895
at endpoints When the dam deforms upstream 119878119894119895will rise
with the rise of 120583119894119895and it will have its maximum value 1198781
119894119895and
minimum value 1198780119894119895at endpoints
The expectation of one deformationmonitoring sequencesample at one observation point (119864
119909lt 0 119878
119894119895) changes with
01
05
Dow
nstre
amU
pstre
am d
ispla
cem
ent
Sij
1205830
1 minus 1205830 120583
disp
lace
men
t
Figure 5 Diagram of 119878119894119895changes with the change of 120583
119894119895if 119864119909lt 0
the change of 120583119894119895as shown in Figure 5 where 120583
0= int0
minusinfin
119891(120589)119889120589
and Figure 5 is presented for 119878119894119895changes with the change of
120583119894119895When the damdeforms downstream the change law of 119878
119894119895
is as follows 119878119894119895will increase with decreasing 120583
119894119895and will have
its maximum value 1198781119894119895and minimum value 1198780
119894119895at endpoints
When the damdeforms upstream the figure shows that when120583119894119895is in the range of (1minus120583
0 05) 119878
119894119895will increase with the drop
of 120583119894119895 when 120583
119894119895is in the range of (05 1) 119878
119894119895will decrease with
the increase of 120583119894119895 when 120583
119894119895= 05 119878
119894119895will reach theminimum
value If 05 is contained in the range of [1205830119894119895 1205831
119894119895] 119878119894119895will have
its minimum value 1198780119894119895at 120583119894119895= 05 and have its maximum
value 1198781119894119895at endpoint if 05 is not contained in the range 119878
119894119895
will have its maximum value 1198781119894119895and minimum value 1198780
119894119895at
endpoints
43 Construction of the Fuzzy Information Entropy of OverallDeformation On the basis of the above results the expres-sion of information entropy of overall deformation can bededucedThe contribution of the order degree of observationpoint 119894 is 120596
119894120583119894119895 make 1205831
119894119895= 120583119894119895and 1205832
119894119895= 1 minus 120583
119894119895 According to
the broad definition of information entropy when the dammove deforms downstream the expression of informationentropy of overall deformation is expressed as follows
119878119895= minus
119898
sum
119894=1
2
sum
119896=1
120596119894120583119896
119894119895ln (120596119894120583119896
119894119895)
= minus
119899
sum
119894=1
2
sum
119896=1
120596119894120583119896
119894119895(ln120596119894+ ln 120583119896
119894119895)
= minus
119899
sum
119894=1
2
sum
119896=1
120596119894120583119896
119894119895ln120596119894minus
119899
sum
119894=1
2
sum
119896=1
120596119894120583119896
119894119895ln 120583119896119894119895
6 Mathematical Problems in Engineering
Choice of sample
Input sample eigenvalues
Calling cloud model program
Generating range of cloud fall
Calculating the fuzzinessof all measured value
Constructing deformed entropyof one measured value
Calculating weight of singleobservation point
Constructing fuzzy deformedentropy of overall deformation
Constructing fuzzy deformedentropy of one measured value
Ex EnHe
Figure 6 Computational process of fuzzy information entropy of overall deformation
= minus
119899
sum
119894=1
120596119894ln120596119894
2
sum
119896=1
120583119896
119894119895minus
119899
sum
119894=1
120596119894
2
sum
119896=1
120583119896
119894119895ln 120583119896119894119895
= minus
119899
sum
119894=1
120596119894ln120596119894+
119899
sum
119894=1
120596119894119878119894119895
(15)
When the dam move deforms upstream the expressionof the information entropy of overall deformation is
119878119895=
119898
sum
119894=1
2
sum
119896=1
120596119894120583119896
119894119895ln (120596119894120583119896
119894119895) =
119899
sum
119894=1
120596119894ln120596119894+
119899
sum
119894=1
120596119894119878119894119895 (16)
Therefore the expression of information entropy of over-all deformation is defined as follows
119878119895=
minus
119899
sum
119894=1
120596119894ln120596119894+
119899
sum
119894=1
120596119894119878119894119895 119878119894119895ge 0
119899
sum
119894=1
120596119894ln120596119894+
119899
sum
119894=1
120596119894119878119894119895 119878
119894119895lt 0
(17)
The absolute value of the information entropy of the over-all deformationmeasures the danger level of the deformationthat is a smaller absolute value means a higher danger levelpositive and negative values stand for the direction of thedeformation a positive value means downstream deforma-tion whereas a negative value means upstream deformation
Considering the influences of random factors the fuzzyinformation entropy of 119909
119894119895is [1198780119894119895 1198781
119894119895] the fuzzy information
entropy of overall deformation can be illustrated through(17) Computational process of fuzzy information entropy ofoverall deformation is shown in Figure 6
5 Proposed Multistage FuzzyInformation Entropy of OverallDeformation Warning Indicators
Horizontal displacement of dam crest changes in an annualcycle ldquoupstream and downstream switchrdquo Therefore thisdisplacement should be in a certain scale and be controlledunder somemonitoring indicators for the safe damoperation
In the case of downstream displacement the primaryfuzzy warning indicator 1205751015840
1is defined as 1205751015840
1= (1205750
1 1205751
1) 12057501is
the lower limit of this indicator and 12057511is the upper limit
the secondary indicator 12057510158402is 12057510158402= (1205750
2 1205751
2) where 1205750
2is
the lower limit of this indicator and 12057512is the upper limit
When 12057511gt 1205750
2 a cross phenomenon appears in the primary
indicator and secondary indicator when both of them shouldbe categorized according to the membership of displacementmeasured 120575lowast was introduced because the membership ofdisplacement at this point is the same Figure 7 showsdiagram of multistage fuzzy information entropy warningindicators
The primary fuzzy warning indicator is 12057510158401= (1205750
1 120575lowast
) andthe secondary is 1205751015840
2= (120575lowast
1205751
2)
If the deformation value is in (12057501 1205751
1) or (1205750
1 120575lowast
) the damis in the state of primary warning if the value is in (1205750
2 1205751
2) or
(1205750
2 120575lowast
) the dam is in the state of secondary warningThe time sequence of deformation at each observation
point was analyzed by using the above theoretical methodThe lower and upper limits of the fuzzy information entropyof overall deformation affected by Δ
119894119895will be 1198780
119895 and 1198781
119895
Considering the damrsquos long-term service when the dammoves downstream the lower limit 1198780
119898119895 and upper limit
Mathematical Problems in Engineering 7
1Degree of membership
Displacement
1Degree of membership
Displacement
12057512120575021205751112057501
1205751212057502 1205751112057501 120575lowast
Figure 7Diagramofmultistage fuzzy information entropywarningindicators
1198781
119898119895 were selected and when the dam moves upstream
the lower limit 1198770119898119895 and upper limit 1198771
119898119895 were selected
1198780
119898119895 1198781119898119895 1198770119898119895 and 1198771
119898119895 are random variables and four
subsample spaces with the sample size of 119873 can be obtainedby the following
1198780
= 1198780
1198981 1198780
1198982 119878
0
119898119898
1198781
= 1198781
1198981 1198781
1198982 119878
1
119898119898
1198770
= 1198770
1198981 1198770
1198982 119877
0
119898119898
1198771
= 1198771
1198981 1198771
1198982 119877
1
119898119898
(18)
Shapiro-Wilk test andKolmogorov-Smirnov test can bothtest whether the samples obey normal distribution or not Butthe Kolmogorov-Smirnov test is applicable to fewer samplesIt can not only test if the samples are subject to normaldistribution but also test if samples are subject to otherdistributions The basic idea of the K-S test is to comparethe cumulative frequency of the observed value (119865
119899(119909)) with
the assumed theoretical probability distribution (119865119909(119909)) to
construct statisticsAccording to the method of empirical distribution func-
tion segmented cumulative frequency is obtained by usingthe following formula
119865119899(119909) =
0 119909 lt 119909119894
119894
119899 119909119894le 119909 lt 119909
119894+1
1 119909 ge 119909
(19)
In the formula 1199091 1199092 119909
119899is sample data after arrange-
ment The sample size is 119899
In the full range of random variable 119883 the maximumdifference between 119865
119899(119909) and 119865
119909(119909) is
119863119899= max 1003816100381610038161003816119865119909 (119909) minus 119865119899 (119909)
1003816100381610038161003816 lt 119863120573
119899 (20)
In the formula 119863119899is a random variable whose distribu-
tion depends on 119899119863120573119899is critical value for a significant level 120573
It is considered that the distribution to be used at a significantlevel 120573 cannot be resisted otherwise it should be rejected
The distribution formwas tested through the K-Smethodto determine the probability density function Fuzzy warningindicator was then determinedwith different significant levelIn dam safety evaluation significant level 120572 is the probabilityof the dam failure Supposing 119878
119898is the extreme of the
information entropy of the upstream overall deformation if119878 gt 119878
119898 the probability of the dam failure is 119875(119878 gt 119878
119898) =
120572 = intinfin
119878119898
119891(119909)119889119909 and the reliability index of dam failure is1 minus 120572 According to the dam importance different failureprobability is set and the multistage warning indicators wereidentified
6 Example Analysis
61 Project Profile One flat-slab deck dam built with rein-forced concrete is an important part of one river basin cascadeexploitation The elevation of this dam crest is 13770m andthe height of biggest part is about 43m the crest runs 2250min length and is made of 27 flat-slab buttresses with a span of75m The space between the left side of 2 buttress and theright side of 9 buttress is the joint part the overflow buttressis located from the 9 buttress to the 20 buttress the rest isthe water-retaining buttressTheworkshop buttress is locatedfrom the 5 buttress to the 8 buttress In this dam the levelof deadwater is 1220m the normal highwater level is 1310m(in practice it is 1290m) the design flood level is 1367m andthe check flood level is 1375m To monitor the displacementof this dam a direct plumb line and an inverted plumb linewere arranged in four buttresses 4 9 21 and 24There isa crushed zone under the dam foundation where occurrenceis N20∘sim25∘W SWang70∘sim80∘ the maximum width is about3m and the narrowest place is about 1mThere is an elevationclip joint mud at level 91m
The deformation field characterized by the observationpoint at 21 should be typical and is a key point because it isin the riverbedThus the observation point G21 at the heightof 134m along the direct plumb line at 21 was analyzed aswell as point 27 at the height of 118m along this line andpoint 28 at the height of 107mAll these valuesmeasuredweretransformed into absolute displacement The arrangement ofeach point is shown in Figure 8 The daily monitoring dataseries is from January 1 2003 to December 31 2013
In this paper two-stage warning indicators were setaccording to the practical running of this project and danger120572 = 5 is the primary warning that is mainly used to discri-minate and handle early dangerous case and the reliabilityindex of dam failure is 95 whereas 120572 = 1 is the secondarywarning that is mainly used to determine grave danger andprevent urgent danger and the reliability index of dam failureis 99
8 Mathematical Problems in Engineering
G21 G9 G4G24
25
30
2631
28
27
3233
Figure 8 The arrangement of observation point
Ups
tream
wat
er le
vel
623
198
4
121
419
89
66
1995
112
620
00
519
200
6
119
201
1
11
1979
Date
120124128132
Figure 9 The process line of upstream water level
11
2007
11
2008
11
2006
123
120
08
11
2005
123
120
09
123
120
10
123
120
11
123
020
12
123
020
13
Data
05
101520253035
Tem
pera
ture
Figure 10 The process line of temperature
62 Calculating the Contribution of Deformation at theObservation Point to the Overall Deformation Figures 9-10 show the upstream stage hydrograph and temperaturestage hydrograph respectively The water stage remainedunchanged whereas the temperature changed in the annualcycle Figure 11 shows the relationship between informationentropy of overall deformation and temperature in 2007 Anegative correlation exists between the overall deformationof 21 buttress and temperature that is an increase in tem-perature corresponds to the decrease in upstream or down-stream displacement and a decrease in temperature meansan increase in the upstream or downstream displacementFigure 12 shows the correlation between the displacementat observation point G21 and the temperature The overalldeformation law is roughly identical with that at observationpoint G21
Figure 12 reveals that the temperature obviously influ-enced overall deformation that is the temperature canchange the contribution of single observation point to theoverall deformation Temperature change was divided intothe stage of temperature rise and stage of temperature drop
2011
11
2011
21
2011
31
2011
41
2011
51
2011
61
2011
71
2011
81
2011
91
2011
10
1
2011
11
1
2011
12
1
TemperatureDeformed entropy
0070
140210280
Tem
pera
ture
minus080minus076minus072minus068minus064minus060
Def
orm
ed en
tropy
Figure 11 The relationship between information entropy of overalldeformation and temperature in 2011
2011
11
2011
13
1
2011
32
2011
41
2011
51
2011
53
1
2011
63
0
2011
73
0
2011
82
9
2011
92
8
2011
10
28
2011
11
27
2011
12
27
Horizontal displacementTemperature
05
101520253035
Tem
pera
ture
minus16minus12minus08minus04004
Hor
izon
tal
disp
lace
men
t
Figure 12 The relationship between horizontal displacement andtemperature of observation point G21 in 2011
The weight of deformation at observation point in two stageswas calculated The results are shown in the Table 1
63 Results of Information Entropy of Overall DeformationRange of cloud fall of the displacement at G21 27 and28 is shown in Figure 13 The boundary values of mostdangerous information entropy from 2003 to 2013 are shownin Tables 2ndash5The absolute value means the degree of dangerdownstream is set as negative value In the significance level120572 = 005 7 kinds of common distribution (lognormal distri-bution normal distribution uniform distribution triangulardistribution exponential distribution 120574 distribution and 120573distribution) were used to carry on hypothesis test for 1198780
119898119895
1198781
119898119895 1198770119898119895 and 1198771
119898119895 with K-S method The maximum 119863
119899
of each sequencewas obtained and comparedwith the criticalvalue 119863005
119899of the significant level 005 to determine the type
Mathematical Problems in Engineering 9
Table 1 The weight table of observation point
Observation point Altitude The period of temperature rise The period of temperature dropG21 134m 0357 039227 118m 0331 031628 107m 0312 0292
Table 2 The lower limit of the most dangerous information entropy of downstream overall deformation
Year 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013Information entropy 04784 04837 04767 04784 04969 04887 04969 05102 05082 04799 04739
Table 3 The upper limit of the most dangerous information entropy of downstream overall deformation
Year 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013Information entropy 04791 04845 04770 04786 04971 04899 04974 05186 05161 04817 04769
Table 4 The lower limit of the most dangerous information entropy of upstream overall deformation
Year 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013Information entropy minus05433 minus05491 minus04773 minus05083 minus05067 minus04784 minus05080 minus04839 minus04836 minus04825 minus05020
Table 5 The lower limit of the most dangerous information entropy of upstream overall deformation
Year 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013Information entropy minus05339 minus05440 minus04770 minus05079 minus05055 minus04770 minus05040 minus04822 minus04806 minus04754 minus04766
Cloud droplets
minus15
minus14
minus13
minus12
minus11
minus1
minus09
minus08
minus07
minus06
minus05
minus04
minus03
minus02
minus01
010
02
03
04
05
06
07
08
09 1
00102030405060708091
Cer
tain
ty
G21 27 28
Figure 13 Range of cloud fall of the displacement at G21 27 and 28
of the best distribution K-S test results are shown in Table 6Multistage fuzzy warning values are presented in Table 7
K-S test shows that 1198780119898119895 1198781119898119895 1198770119898119895 and 1198771
119898119895 satisfy
normal distributionThe probability density function of the sequence is
119891 (119909) =1
radic21205871205902exp(minus
(119909 minus 120583)2
120590) (21)
Parameter values of (21) are presented in Table 7In downstream deformation if 120572 = 5 the primary
warning indicator is (04763 04775) if 120572 = 1 thesecondary warning indicator is (04757 04763) 120575lowast is used as
the boundary value when two indicators overlap In the caseof upstream deformation if 120572 = 5 the primary warningindicator is (minus04779 minus04768) if 120572 = 1 the secondarywarning indicator is (minus04768 minus04763) (Table 8)
64 The Structure Calculation of Monitoring Index of DamHorizontal Displacement According to the actual situationthree-dimensional finite element model of the dam is estab-lished According to the structure and basic geologicalconditions of 21 dam section the scope of finite elementcalculation model can be got as follows taking 15 times ashigh dam in the upstream direction taking 15 times as highdam in the upstream direction and taking 1 time as highdam below the dam foundation The model is constitutedof 11445 nodes and 8571 units The unit type is 6 sides 8nodes isoparametric element The deformation observationdata analysis shows the dam under the condition of lowtemperature and highwater level there is larger displacementin the downstream when in high temperature and in lowwater level there is larger displacement in the upstream(Table 9 Figure 14) In view of the actual situation of thedam observation and data analysis results the load conditionselection is as follows (Table 10 Figures 15 and 16)
Working condition 1 normal water level 1290m andmaximum temperature dropWorking condition 2 dead water level 1220m andmaximum temperature rise
10 Mathematical Problems in Engineering
Table 6 K-S test results
Probability distribution 1198780
1198981198951198781
1198981198951198770
1198981198951198771
119898119895
Lognormal distribution 026 011 023 028Normal distribution 011 008 026 022Uniform distribution 074 155 088 068Triangular distribution 053 054 059 061Exponential distribution 041 042 055 035120574 distribution 037 036 031 033120573 distribution 068 073 086 063119863005
119899034 034 029 029
The most reasonable probability distribution Normal distribution Normal distribution Normal distribution Normal distribution
Table 7 Parameter values of the probability density function
Data series Parameter values120583 120590
2
1198780
119898119895 0488355 00129
1198781
119898119895 0490627 00151
1198770
119898119895 minus050210 00249
1198771
119898119895 minus049674 00245
Figure 14 Finite element model of the dam
The primary warning indicators of concrete dam wereobtained by calculation methods for structures If the infor-mation entropy of the overall deformation reached 04770the dammoveddownstreamand in the state of primarywarn-ing If the information entropy of the overall deformationreached minus04773 the dam moved upstream and in the stateof primary warning (Table 11) They all fell into intervalscalculated by fuzzy methodologyThe analysis shows that themethod brought up in this paper is reasonable and scientificAlso the analysis shows the physical meaning of the fuzzywarning indexUnder the action of the unfavorable load com-bination and the influence of the complex random factorsthe maximum entropy andminimum information entropy ofthe overall deformation lie in this interval Considering theinfluences of random factorsmultistage fuzzywarning valueswere receptive and safe
In this paper the overall deformation of 21 buttresswas analyzed through the theoretical method proposedInfluences of random factors on the warning value wereconsidered and multistage fuzzy warning values were deter-mined If the information entropy of overall deformationis in (04763 04775) the dam moved downstream and in
minus2399e minus 003minus1296e minus 003minus1940e minus 0049084e minus 0042011e minus 0033113e minus 0034216e minus 0035318e minus 0036421e minus 0037523e minus 0038625e minus 003
XY
Z
Figure 15The results of finite element calculation ofworking condi-tion 1
XY
Z minus6007e minus 004minus5014e minus 004minus4021e minus 004minus3028e minus 004minus2035e minus 004minus1042e minus 004minus4872e minus 0069443e minus 0051937e minus 0042930e minus 0043923e minus 004
Figure 16 The results of finite element calculation of workingcondition 2
the state of primary warning If the information entropy ofoverall deformation is in (04757 04763) the dam moveddownstream and in the state of secondary warning If theinformation entropy of overall deformation is in (minus04779minus04768) the dam moved upstream and in the state ofprimary warning If the information entropy of the over-all deformation is in (minus04768 minus04763) the dam movedupstream and in the state of secondary warning
7 Conclusion
This paper presented multistage warning indicators of con-crete dam space and considered the influences of complex
Mathematical Problems in Engineering 11
Table 8 Multistage fuzzy warning values of concrete dam
The direction of deformation Confidence levelThe primary warning indicator The secondary warning indicator
Downstream (04763 04775) (04757 04763)Upstream (minus04779 minus04768) (minus04768 minus04763)
Table 9 Material parameter of the dam
Structure Density (kgm) Poissonrsquos ratio Elastic modulus (GPa)Concrete face slab 2400 0167 24Buttress 2400 0167 24Partition wall 2400 0167 24Reinforced concrete block 2400 0160 22Foundation rock mass 2700 0175 12Diorite-dyke 2000 03 115Crushed zone 2000 03 029Horizontal joints 2000 03 07
Table 10 The results of finite element calculation of displacementof observation point (mm)
Observation point Working condition 1 Working condition 2G21 0785 minus059927 0654 minus038628 0356 0114
Table 11 The primary warning indicators of concrete dam
The direction of deformation The primary warning indicatorDownstream 04770Upstream minus04773
random factors The results of the specific studies are asfollows
(1) Influences of fuzziness and randomness of randomfactors on the long-term service of dam were dis-cussed a fuzziness indicator that measures the influ-ence of random factors on monitoring value wasconstructed through cloud model
(2) Equivalent model of overall deformation was pro-posed In the model the overall deformation ofdam was regarded as a deformation system whereeach observation point had different contributions(weights) and affected one another Based on entropytheory a space information entropy that can measurethe overall deformation condition was established
(3) Multistage warning indicators against overall defor-mation of concrete dam under the influences of fuzzi-ness and randomness were determined and nondeter-ministic optimal control of the indicator was achievedto improve the competence of warning against thedeformation of concrete dam
Competing Interests
No conflict of interests exits in the submission of this paper
Acknowledgments
This paper was financially supported by National Natu-ral Science Foundation of China (Grants nos 4132300151139001 51579086 51379068 51279052 51579083 and51209077) Jiangsu Natural Science Foundation (Grants nosBK20140039 and BK2012036) Research Fund for the Doc-toral Program of Higher Education of China (Grant no20130094110010) the Ministry of Water Resources PublicWelfare Industry Research Special Fund Project (Grantsnos 201201038 and 201301061) Jiangsu Province ldquo333 High-Level Personnel Training Projectrdquo (Grants nos BRA2011179and BRA2011145) Project Funded by the Priority AcademicProgram Development of Jiangsu Higher Education Institu-tions (Grant no YS11001) Jiangsu Province ldquo333 High-LevelPersonnel Training Projectrdquo (Grant no 2017-B08037) JiangsuProvince ldquoSix Talent Peaksrdquo Project (Grant no JY-008) andFundamental Research Funds for the Central Universities(Grants nos 2016B04114 and 2015B25414)
References
[1] F BenedettoGGiunta andLMastroeni ldquoAmaximumentropymethod to assess the predictability of financial and commoditypricesrdquo Digital Signal Processing vol 46 pp 19ndash31 2015
[2] S Muraro A Madaschi and A Gajo ldquoOn the reliability of 3Dnumerical analyses on passive piles used for slope stabilisationin frictional soilsrdquo Geotechnique vol 64 no 6 pp 486ndash4922014
[3] W J Yoon and K S Park ldquoA study on the market instabilityindex and risk warning levels in early warning system foreconomic crisisrdquo Digital Signal Processing vol 29 pp 35ndash442014
[4] D Tonini ldquoObserved behavior of several leakier arch damsrdquoJournal of the Power Division vol 82 no 12 pp 135ndash139 1956
12 Mathematical Problems in Engineering
[5] R Salgado and D Kim ldquoReliability analysis of load andresistance factor design of slopesrdquo Journal of Geotechnical andGeoenvironmental Engineering vol 140 no 1 pp 57ndash73 2014
[6] Y C Gu and S J Wang ldquoStudy on the early-warning thresholdof structural instability unexpected accidents of reservoir damrdquoJournal of Hydraulic Engineering vol 40 no 12 pp 1467ndash14722009
[7] H Z Su F Wang and H P Liu ldquoEarly-warning index for damservice behavior based on POT modelrdquo Journal of HydraulicEngineering vol 43 no 8 pp 974ndash986 2012
[8] L Chen Mechanics performance with parameters changing inspace amp monitoring model of roller compacted concrete dam[PhD thesis] Hohai University 2006
[9] A B Li L F Zhang Q L Wang B Li Z Li and Y WangldquoInformation theory in nonlinear error growth dynamics andits application to predictability taking the Lorenz system as anexamplerdquo Science China Earth Sciences vol 56 no 8 pp 1413ndash1421 2013
[10] O Victor ldquoThe theory of cloudsrdquo Library Journal vol 132 no13 p 62 2007
[11] M Yang H Z Su and X Q Yan ldquoComputation and analysis ofhigh rocky slope safety in a water conservancy projectrdquoDiscreteDynamics in Nature and Society vol 2015 Article ID 197579 11pages 2015
[12] C Ricotta and G C Avena ldquoEvaluating the degree of fuzzi-ness of thematic maps with a generalized entropy functiona methodological outlookrdquo International Journal of RemoteSensing vol 23 no 20 pp 4519ndash4523 2002
[13] V I Khvorostyanov and J A Curry ldquoParameterization ofhomogeneous ice nucleation for cloud and climate modelsbased on classical nucleation theoryrdquo Atmospheric Chemistryand Physics vol 12 no 19 pp 9275ndash9302 2012
[14] H Z Su Z P Wen and Z R Wu ldquoStudy on an intelligentinference engine in early-warning system of damhealthrdquoWaterResources Management vol 25 no 6 pp 1545ndash1563 2011
[15] M Kaminski ldquoProbabilistic entropy in homogenization ofthe periodic fiber-reinforced composites with random elasticparametersrdquo International Journal for Numerical Methods inEngineering vol 90 no 8 pp 939ndash954 2012
[16] E Czogała and J Łeski ldquoApplication of entropy and energymeasures of fuzziness to processing of ECG signalrdquo Fuzzy Setsand Systems vol 97 no 1 pp 9ndash18 1998
[17] K Wu and J L Jin ldquoProjection pursuit model for evaluation ofregion water resource security based on changeable weight andinformationrdquo Resources and Environment in the Yangtze Basinvol 20 no 9 pp 1085ndash1091 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
War
ning
val
ue o
f dow
nstre
am
disp
lace
men
t
War
ning
val
ue o
f ups
tream
di
spla
cem
ent
Displacement
Probability density
Displacement
The m
axim
um co
ntro
l val
ue o
f war
ning
va
lue o
f dow
nstre
am d
ispla
cem
ent
The m
inim
um co
ntro
l val
ue o
f war
ning
va
lue o
f dow
nstre
am d
ispla
cem
ent
The m
axim
um co
ntro
l val
ue o
f war
ning
va
lue o
f ups
tream
disp
lace
men
t
The m
inim
um co
ntro
l val
ue o
f war
ning
va
lue o
f ups
tream
disp
lace
men
t
Probability density
Figure 1 Diagram of deterministic optimal control and nondeterministic optimal control
Degree of certainty
Upper limit of cloud dropletsLower limit of cloud droplets
Cloud dropletsEx
Figure 2 Diagram of the range where the cloud droplet falls
known the corresponding information entropy sequence canbe calculated Deformation of dam can be divided into threeparts water pressure component temperature componentand aging component Aging component comprehensivelyreflects the creep and plastic deformation of dam concreteand rock foundation and compression deformation of geo-logical structure of rock foundation At the same time italso includes the irreversible displacement caused by the damcrack and the autogenous volume deformation It changesdramatically in the early stage and gradually tends to be stablein the later stage The project selected in this paper is a damwhich has worked for many years and its aging componentstend to be stable and the deformation value is stable in annualperiod rule which obeys normal distribution
The indicator measuring how much information is con-tained in 119909
119894119895is the inverse entropy 119863
119894119895 119863119894119895= 1 minus 119878
119894119895
When one measured value reflects more information theentropy value 119878
119894119895will be smaller and its inverse entropy 119863
119894119895
will be greater Thus 119863119894119895can be used to measure how much
information is reflected by a single measured value
Suppose the weight distribution of all observation pointsis 120596119894| 119894 = 1 2 119899 where 119899 is the number of points
observed and 120596119894will meet the following requirement 120596
119894ge 0
and sum120596119894= 1 The entropy 119878
119894119895of the object matrix of the
deformation measured value 119909119894119895| 119894 = 1 sim 119899 119895 = 1 sim 119898
can be computed through (4) and (5) The inverse entropymatrix is expressed as follows
119863119894119895=
[[[[[[
[
1198631111986312sdot sdot sdot 119863
1119898
1198632111986322sdot sdot sdot 119863
2119898
11986311989911198631198992sdot sdot sdot 119863
119899119898
]]]]]]
]
(6)
In this matrix 119863119894119895is the inverse entropy of 119909
119894119895 The weight
of the feature points was traced by the projection pursuitmethod
32 Process of Calculating the Weight of a Single Obser-vation Point By using the projection pursuit method [1617] the high-dimensional data can be projected to lowdimension space and projection that reflects the structureor features of high-dimensional data is pursued to analyzehigh-dimensional dataThismethod is advantageous becauseit is highly objective robust resistant to interference andaccurate The steps are as follows
Step 1 The extreme value of the inverse entropy matrix wasnormalized through the following equation
119863lowast
119894119895=
119863119894119895minus [119863119895]min
[119863119895]max minus [119863119895]min
(7)
where [119863119895]max and [119863119895]min are the maximum and minimum
values of line 119895 in the matrix respectively
4 Mathematical Problems in Engineering
Weight of surveypoints 1
Overall deformation of concrete dam
Deformation ofsurvey points 1
Deformation ofsurvey points n
Deformation ofsurvey points 2
Weight of surveypoints 2
Weight of surveypoints n
middot middot middot
middot middot middot
Figure 3 Diagram of overall deformation system of concrete dam
Step 2 The normalized value 119863lowast119894119895
was projected to unitdirection119875 119875 = (119901
1 1199012 119901
119895) and1199012
1+1199012
2+sdot sdot sdot+119901
2
119895= 1The
indicator function of the projection 119866(119894) was constructed
119866 (119894) =
119898
sum
119895=1
119901119895119863lowast
119894119895 (119894 = 1 2 119899) (8)
Step 3 The objective function of the projection was con-structed The best direction for the projection was estimatedby solving the maximization problem of the objective func-tion in the constraint condition
Objective function Max 119867 (119901) = 119878119866sdot 119876119866
Constraint condition119898
sum
119895=1
1199012
119895= 1
(9)
In this equation 119878119866is the divergent degree of the projection
119876119866is the local density of 1D data points along 119875 and is
expressed as follows
119878119866= [sum119899
119894=1(119866 (119894) minus 119892 (119894))
2
119899 minus 1]
05
119876119866=
119899
sum
119894=1
119899
sum
119895=1
(119877 minus 119903119894119895) sdot 119891 (119877 minus 119903
119894119895)
(10)
where119892(119894) is themean value of this sequence119877 is the windowradius of the local density and 119877 = 01119878
119866in this paper 119903
119894119895is
the distance between two projection values and 119891(119905) is theunit step function 119891(119905) is equal to 1 as 119905 is greater than 0Otherwise 119891(119905) is equal to 0
Step 4 The projection value of one sample point was com-puted by substituting the best direction 119875lowast into (8) 120596
119894can be
calculated by normalizing the projection value
120596119894=
119866lowast
(119894)
sum119899
119895=1119866lowast (119895)
119894 = 1 2 119899 (11)
4 Study on Equivalent Model ofDamrsquos Overall Deformation
The overall deformation of a dam can be considered a defor-mation system of feature points with different contributionsthat influence one another as well as observation points ofthe deformation as feature pointsThe deformation conditionwas analyzed systemically and the overall deformation wasexpressed by the evolution of equation of all feature pointsThe deformation condition was described qualitatively bythe tectonic type of information entropy The absolute valueof the information entropy measures the danger level ofthe deformation value Smaller absolute value means higherdanger level The positive and negative values indicate thedirection of the deformation A positive value correspondsto downstream deformation whereas a negative value corre-sponds to upstream deformation The influences of randomfactors were considered and the fuzzy information entropywas constructed
41 Constructing Fuzzy Information Entropy of SingleMeasured Value The downstream deformation is positivewhereas the upstream deformation is negative
When the observation point moves downstream make120583119894119895= int119909119894119895
minusinfin
119891(120589)119889120589 and according to the definition of infor-mation entropy the information entropy of 119909
119894119895is expressed
as (4)Considering the influence of Δ
119894119895 120583119894119895will float in [1205830
119894119895 1205831
119894119895]
the following is then obtained
1205830
119894119895= int
119909119894119895minus119909119894119895Δ 119894119895
minusinfin
119891 (120589) 119889120589
1205831
119894119895= int
119909119894119895+119909119894119895Δ 119894119895
minusinfin
119891 (120589) 119889120589
(12)
When the observation point moves upstream make 120583119894119895=
int+infin
119909119894119895
119891(120589)119889120589 the information entropy of 119909119894119895is defined as
follows119878119894119895= 120583119894119895ln 120583119894119895+ (1 minus 120583
119894119895) ln (1 minus 120583
119894119895) (13)
Mathematical Problems in Engineering 5
0
Dow
nstre
am d
ispla
cem
ent
Ups
tream
di
spla
cem
ent
105
Sij
1205830
1 minus 1205830 120583
Figure 4 Diagram of 119878119894119895changes with the change of 120583
119894119895if 119864119909ge 0
Considering the influence ofΔ119894119895 120583119894119895floats in [1205830
119894119895 1205831
119894119895]The
following is then obtained
1205830
119894119895= int
+infin
119909119894119895minus119909119894119895Δ 119894119895
119891 (120589) 119889120589
1205831
119894119895= int
+infin
119909119894119895+119909119894119895Δ 119894119895
119891 (120589) 119889120589
(14)
Considering the influences of random factors the infor-mation entropy of 119909
119894119895minus119878119894119895floats in [1198780
119894119895 1198781
119894119895] whichwas defined
as the fuzzy information entropy of 119909119894119895
Take the downstream as an example Influenced by deter-mined and random factors the fuzzy entropy of 119878
119894119895changes
into the range of 119878119894119895in the range of [1205830
119894119895 1205831
119894119895]
42 Methods for Determining the Range of the InformationEntropy of Single Measured Value For Figure 4 the expecta-tion of one deformation monitoring sequence sample at oneobservation point (119864
119909ge 0 119878
119894119895) changes with the change of
120583119894119895 as shown in Figure 4 where 120583
0= int0
minusinfin
119891(120589)119889120589When the dam deforms downstream the change law of
119878119894119895is as follows 119878
119894119895will increase with increasing 120583
119894119895when 120583
119894119895
is in the range of (1205830 05) 119878
119894119895will decrease with decreasing
120583119894119895when 120583
119894119895is in the range of (05 1) when 120583
119894119895= 05 119878
119894119895will
reach themaximumvalue If 05 is in the range of [1205830119894119895 1205831
119894119895] the
maximum of 119878119894119895is 1198781119894119895when 120583
119894119895= 05 and its minimum value
is 1198780119894119895at the endpoint If 05 is not in the range of [1205830
119894119895 1205831
119894119895]
119878119894119895will have its maximum value 1198781
119894119895and minimum value 1198780
119894119895
at endpoints When the dam deforms upstream 119878119894119895will rise
with the rise of 120583119894119895and it will have its maximum value 1198781
119894119895and
minimum value 1198780119894119895at endpoints
The expectation of one deformationmonitoring sequencesample at one observation point (119864
119909lt 0 119878
119894119895) changes with
01
05
Dow
nstre
amU
pstre
am d
ispla
cem
ent
Sij
1205830
1 minus 1205830 120583
disp
lace
men
t
Figure 5 Diagram of 119878119894119895changes with the change of 120583
119894119895if 119864119909lt 0
the change of 120583119894119895as shown in Figure 5 where 120583
0= int0
minusinfin
119891(120589)119889120589
and Figure 5 is presented for 119878119894119895changes with the change of
120583119894119895When the damdeforms downstream the change law of 119878
119894119895
is as follows 119878119894119895will increase with decreasing 120583
119894119895and will have
its maximum value 1198781119894119895and minimum value 1198780
119894119895at endpoints
When the damdeforms upstream the figure shows that when120583119894119895is in the range of (1minus120583
0 05) 119878
119894119895will increase with the drop
of 120583119894119895 when 120583
119894119895is in the range of (05 1) 119878
119894119895will decrease with
the increase of 120583119894119895 when 120583
119894119895= 05 119878
119894119895will reach theminimum
value If 05 is contained in the range of [1205830119894119895 1205831
119894119895] 119878119894119895will have
its minimum value 1198780119894119895at 120583119894119895= 05 and have its maximum
value 1198781119894119895at endpoint if 05 is not contained in the range 119878
119894119895
will have its maximum value 1198781119894119895and minimum value 1198780
119894119895at
endpoints
43 Construction of the Fuzzy Information Entropy of OverallDeformation On the basis of the above results the expres-sion of information entropy of overall deformation can bededucedThe contribution of the order degree of observationpoint 119894 is 120596
119894120583119894119895 make 1205831
119894119895= 120583119894119895and 1205832
119894119895= 1 minus 120583
119894119895 According to
the broad definition of information entropy when the dammove deforms downstream the expression of informationentropy of overall deformation is expressed as follows
119878119895= minus
119898
sum
119894=1
2
sum
119896=1
120596119894120583119896
119894119895ln (120596119894120583119896
119894119895)
= minus
119899
sum
119894=1
2
sum
119896=1
120596119894120583119896
119894119895(ln120596119894+ ln 120583119896
119894119895)
= minus
119899
sum
119894=1
2
sum
119896=1
120596119894120583119896
119894119895ln120596119894minus
119899
sum
119894=1
2
sum
119896=1
120596119894120583119896
119894119895ln 120583119896119894119895
6 Mathematical Problems in Engineering
Choice of sample
Input sample eigenvalues
Calling cloud model program
Generating range of cloud fall
Calculating the fuzzinessof all measured value
Constructing deformed entropyof one measured value
Calculating weight of singleobservation point
Constructing fuzzy deformedentropy of overall deformation
Constructing fuzzy deformedentropy of one measured value
Ex EnHe
Figure 6 Computational process of fuzzy information entropy of overall deformation
= minus
119899
sum
119894=1
120596119894ln120596119894
2
sum
119896=1
120583119896
119894119895minus
119899
sum
119894=1
120596119894
2
sum
119896=1
120583119896
119894119895ln 120583119896119894119895
= minus
119899
sum
119894=1
120596119894ln120596119894+
119899
sum
119894=1
120596119894119878119894119895
(15)
When the dam move deforms upstream the expressionof the information entropy of overall deformation is
119878119895=
119898
sum
119894=1
2
sum
119896=1
120596119894120583119896
119894119895ln (120596119894120583119896
119894119895) =
119899
sum
119894=1
120596119894ln120596119894+
119899
sum
119894=1
120596119894119878119894119895 (16)
Therefore the expression of information entropy of over-all deformation is defined as follows
119878119895=
minus
119899
sum
119894=1
120596119894ln120596119894+
119899
sum
119894=1
120596119894119878119894119895 119878119894119895ge 0
119899
sum
119894=1
120596119894ln120596119894+
119899
sum
119894=1
120596119894119878119894119895 119878
119894119895lt 0
(17)
The absolute value of the information entropy of the over-all deformationmeasures the danger level of the deformationthat is a smaller absolute value means a higher danger levelpositive and negative values stand for the direction of thedeformation a positive value means downstream deforma-tion whereas a negative value means upstream deformation
Considering the influences of random factors the fuzzyinformation entropy of 119909
119894119895is [1198780119894119895 1198781
119894119895] the fuzzy information
entropy of overall deformation can be illustrated through(17) Computational process of fuzzy information entropy ofoverall deformation is shown in Figure 6
5 Proposed Multistage FuzzyInformation Entropy of OverallDeformation Warning Indicators
Horizontal displacement of dam crest changes in an annualcycle ldquoupstream and downstream switchrdquo Therefore thisdisplacement should be in a certain scale and be controlledunder somemonitoring indicators for the safe damoperation
In the case of downstream displacement the primaryfuzzy warning indicator 1205751015840
1is defined as 1205751015840
1= (1205750
1 1205751
1) 12057501is
the lower limit of this indicator and 12057511is the upper limit
the secondary indicator 12057510158402is 12057510158402= (1205750
2 1205751
2) where 1205750
2is
the lower limit of this indicator and 12057512is the upper limit
When 12057511gt 1205750
2 a cross phenomenon appears in the primary
indicator and secondary indicator when both of them shouldbe categorized according to the membership of displacementmeasured 120575lowast was introduced because the membership ofdisplacement at this point is the same Figure 7 showsdiagram of multistage fuzzy information entropy warningindicators
The primary fuzzy warning indicator is 12057510158401= (1205750
1 120575lowast
) andthe secondary is 1205751015840
2= (120575lowast
1205751
2)
If the deformation value is in (12057501 1205751
1) or (1205750
1 120575lowast
) the damis in the state of primary warning if the value is in (1205750
2 1205751
2) or
(1205750
2 120575lowast
) the dam is in the state of secondary warningThe time sequence of deformation at each observation
point was analyzed by using the above theoretical methodThe lower and upper limits of the fuzzy information entropyof overall deformation affected by Δ
119894119895will be 1198780
119895 and 1198781
119895
Considering the damrsquos long-term service when the dammoves downstream the lower limit 1198780
119898119895 and upper limit
Mathematical Problems in Engineering 7
1Degree of membership
Displacement
1Degree of membership
Displacement
12057512120575021205751112057501
1205751212057502 1205751112057501 120575lowast
Figure 7Diagramofmultistage fuzzy information entropywarningindicators
1198781
119898119895 were selected and when the dam moves upstream
the lower limit 1198770119898119895 and upper limit 1198771
119898119895 were selected
1198780
119898119895 1198781119898119895 1198770119898119895 and 1198771
119898119895 are random variables and four
subsample spaces with the sample size of 119873 can be obtainedby the following
1198780
= 1198780
1198981 1198780
1198982 119878
0
119898119898
1198781
= 1198781
1198981 1198781
1198982 119878
1
119898119898
1198770
= 1198770
1198981 1198770
1198982 119877
0
119898119898
1198771
= 1198771
1198981 1198771
1198982 119877
1
119898119898
(18)
Shapiro-Wilk test andKolmogorov-Smirnov test can bothtest whether the samples obey normal distribution or not Butthe Kolmogorov-Smirnov test is applicable to fewer samplesIt can not only test if the samples are subject to normaldistribution but also test if samples are subject to otherdistributions The basic idea of the K-S test is to comparethe cumulative frequency of the observed value (119865
119899(119909)) with
the assumed theoretical probability distribution (119865119909(119909)) to
construct statisticsAccording to the method of empirical distribution func-
tion segmented cumulative frequency is obtained by usingthe following formula
119865119899(119909) =
0 119909 lt 119909119894
119894
119899 119909119894le 119909 lt 119909
119894+1
1 119909 ge 119909
(19)
In the formula 1199091 1199092 119909
119899is sample data after arrange-
ment The sample size is 119899
In the full range of random variable 119883 the maximumdifference between 119865
119899(119909) and 119865
119909(119909) is
119863119899= max 1003816100381610038161003816119865119909 (119909) minus 119865119899 (119909)
1003816100381610038161003816 lt 119863120573
119899 (20)
In the formula 119863119899is a random variable whose distribu-
tion depends on 119899119863120573119899is critical value for a significant level 120573
It is considered that the distribution to be used at a significantlevel 120573 cannot be resisted otherwise it should be rejected
The distribution formwas tested through the K-Smethodto determine the probability density function Fuzzy warningindicator was then determinedwith different significant levelIn dam safety evaluation significant level 120572 is the probabilityof the dam failure Supposing 119878
119898is the extreme of the
information entropy of the upstream overall deformation if119878 gt 119878
119898 the probability of the dam failure is 119875(119878 gt 119878
119898) =
120572 = intinfin
119878119898
119891(119909)119889119909 and the reliability index of dam failure is1 minus 120572 According to the dam importance different failureprobability is set and the multistage warning indicators wereidentified
6 Example Analysis
61 Project Profile One flat-slab deck dam built with rein-forced concrete is an important part of one river basin cascadeexploitation The elevation of this dam crest is 13770m andthe height of biggest part is about 43m the crest runs 2250min length and is made of 27 flat-slab buttresses with a span of75m The space between the left side of 2 buttress and theright side of 9 buttress is the joint part the overflow buttressis located from the 9 buttress to the 20 buttress the rest isthe water-retaining buttressTheworkshop buttress is locatedfrom the 5 buttress to the 8 buttress In this dam the levelof deadwater is 1220m the normal highwater level is 1310m(in practice it is 1290m) the design flood level is 1367m andthe check flood level is 1375m To monitor the displacementof this dam a direct plumb line and an inverted plumb linewere arranged in four buttresses 4 9 21 and 24There isa crushed zone under the dam foundation where occurrenceis N20∘sim25∘W SWang70∘sim80∘ the maximum width is about3m and the narrowest place is about 1mThere is an elevationclip joint mud at level 91m
The deformation field characterized by the observationpoint at 21 should be typical and is a key point because it isin the riverbedThus the observation point G21 at the heightof 134m along the direct plumb line at 21 was analyzed aswell as point 27 at the height of 118m along this line andpoint 28 at the height of 107mAll these valuesmeasuredweretransformed into absolute displacement The arrangement ofeach point is shown in Figure 8 The daily monitoring dataseries is from January 1 2003 to December 31 2013
In this paper two-stage warning indicators were setaccording to the practical running of this project and danger120572 = 5 is the primary warning that is mainly used to discri-minate and handle early dangerous case and the reliabilityindex of dam failure is 95 whereas 120572 = 1 is the secondarywarning that is mainly used to determine grave danger andprevent urgent danger and the reliability index of dam failureis 99
8 Mathematical Problems in Engineering
G21 G9 G4G24
25
30
2631
28
27
3233
Figure 8 The arrangement of observation point
Ups
tream
wat
er le
vel
623
198
4
121
419
89
66
1995
112
620
00
519
200
6
119
201
1
11
1979
Date
120124128132
Figure 9 The process line of upstream water level
11
2007
11
2008
11
2006
123
120
08
11
2005
123
120
09
123
120
10
123
120
11
123
020
12
123
020
13
Data
05
101520253035
Tem
pera
ture
Figure 10 The process line of temperature
62 Calculating the Contribution of Deformation at theObservation Point to the Overall Deformation Figures 9-10 show the upstream stage hydrograph and temperaturestage hydrograph respectively The water stage remainedunchanged whereas the temperature changed in the annualcycle Figure 11 shows the relationship between informationentropy of overall deformation and temperature in 2007 Anegative correlation exists between the overall deformationof 21 buttress and temperature that is an increase in tem-perature corresponds to the decrease in upstream or down-stream displacement and a decrease in temperature meansan increase in the upstream or downstream displacementFigure 12 shows the correlation between the displacementat observation point G21 and the temperature The overalldeformation law is roughly identical with that at observationpoint G21
Figure 12 reveals that the temperature obviously influ-enced overall deformation that is the temperature canchange the contribution of single observation point to theoverall deformation Temperature change was divided intothe stage of temperature rise and stage of temperature drop
2011
11
2011
21
2011
31
2011
41
2011
51
2011
61
2011
71
2011
81
2011
91
2011
10
1
2011
11
1
2011
12
1
TemperatureDeformed entropy
0070
140210280
Tem
pera
ture
minus080minus076minus072minus068minus064minus060
Def
orm
ed en
tropy
Figure 11 The relationship between information entropy of overalldeformation and temperature in 2011
2011
11
2011
13
1
2011
32
2011
41
2011
51
2011
53
1
2011
63
0
2011
73
0
2011
82
9
2011
92
8
2011
10
28
2011
11
27
2011
12
27
Horizontal displacementTemperature
05
101520253035
Tem
pera
ture
minus16minus12minus08minus04004
Hor
izon
tal
disp
lace
men
t
Figure 12 The relationship between horizontal displacement andtemperature of observation point G21 in 2011
The weight of deformation at observation point in two stageswas calculated The results are shown in the Table 1
63 Results of Information Entropy of Overall DeformationRange of cloud fall of the displacement at G21 27 and28 is shown in Figure 13 The boundary values of mostdangerous information entropy from 2003 to 2013 are shownin Tables 2ndash5The absolute value means the degree of dangerdownstream is set as negative value In the significance level120572 = 005 7 kinds of common distribution (lognormal distri-bution normal distribution uniform distribution triangulardistribution exponential distribution 120574 distribution and 120573distribution) were used to carry on hypothesis test for 1198780
119898119895
1198781
119898119895 1198770119898119895 and 1198771
119898119895 with K-S method The maximum 119863
119899
of each sequencewas obtained and comparedwith the criticalvalue 119863005
119899of the significant level 005 to determine the type
Mathematical Problems in Engineering 9
Table 1 The weight table of observation point
Observation point Altitude The period of temperature rise The period of temperature dropG21 134m 0357 039227 118m 0331 031628 107m 0312 0292
Table 2 The lower limit of the most dangerous information entropy of downstream overall deformation
Year 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013Information entropy 04784 04837 04767 04784 04969 04887 04969 05102 05082 04799 04739
Table 3 The upper limit of the most dangerous information entropy of downstream overall deformation
Year 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013Information entropy 04791 04845 04770 04786 04971 04899 04974 05186 05161 04817 04769
Table 4 The lower limit of the most dangerous information entropy of upstream overall deformation
Year 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013Information entropy minus05433 minus05491 minus04773 minus05083 minus05067 minus04784 minus05080 minus04839 minus04836 minus04825 minus05020
Table 5 The lower limit of the most dangerous information entropy of upstream overall deformation
Year 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013Information entropy minus05339 minus05440 minus04770 minus05079 minus05055 minus04770 minus05040 minus04822 minus04806 minus04754 minus04766
Cloud droplets
minus15
minus14
minus13
minus12
minus11
minus1
minus09
minus08
minus07
minus06
minus05
minus04
minus03
minus02
minus01
010
02
03
04
05
06
07
08
09 1
00102030405060708091
Cer
tain
ty
G21 27 28
Figure 13 Range of cloud fall of the displacement at G21 27 and 28
of the best distribution K-S test results are shown in Table 6Multistage fuzzy warning values are presented in Table 7
K-S test shows that 1198780119898119895 1198781119898119895 1198770119898119895 and 1198771
119898119895 satisfy
normal distributionThe probability density function of the sequence is
119891 (119909) =1
radic21205871205902exp(minus
(119909 minus 120583)2
120590) (21)
Parameter values of (21) are presented in Table 7In downstream deformation if 120572 = 5 the primary
warning indicator is (04763 04775) if 120572 = 1 thesecondary warning indicator is (04757 04763) 120575lowast is used as
the boundary value when two indicators overlap In the caseof upstream deformation if 120572 = 5 the primary warningindicator is (minus04779 minus04768) if 120572 = 1 the secondarywarning indicator is (minus04768 minus04763) (Table 8)
64 The Structure Calculation of Monitoring Index of DamHorizontal Displacement According to the actual situationthree-dimensional finite element model of the dam is estab-lished According to the structure and basic geologicalconditions of 21 dam section the scope of finite elementcalculation model can be got as follows taking 15 times ashigh dam in the upstream direction taking 15 times as highdam in the upstream direction and taking 1 time as highdam below the dam foundation The model is constitutedof 11445 nodes and 8571 units The unit type is 6 sides 8nodes isoparametric element The deformation observationdata analysis shows the dam under the condition of lowtemperature and highwater level there is larger displacementin the downstream when in high temperature and in lowwater level there is larger displacement in the upstream(Table 9 Figure 14) In view of the actual situation of thedam observation and data analysis results the load conditionselection is as follows (Table 10 Figures 15 and 16)
Working condition 1 normal water level 1290m andmaximum temperature dropWorking condition 2 dead water level 1220m andmaximum temperature rise
10 Mathematical Problems in Engineering
Table 6 K-S test results
Probability distribution 1198780
1198981198951198781
1198981198951198770
1198981198951198771
119898119895
Lognormal distribution 026 011 023 028Normal distribution 011 008 026 022Uniform distribution 074 155 088 068Triangular distribution 053 054 059 061Exponential distribution 041 042 055 035120574 distribution 037 036 031 033120573 distribution 068 073 086 063119863005
119899034 034 029 029
The most reasonable probability distribution Normal distribution Normal distribution Normal distribution Normal distribution
Table 7 Parameter values of the probability density function
Data series Parameter values120583 120590
2
1198780
119898119895 0488355 00129
1198781
119898119895 0490627 00151
1198770
119898119895 minus050210 00249
1198771
119898119895 minus049674 00245
Figure 14 Finite element model of the dam
The primary warning indicators of concrete dam wereobtained by calculation methods for structures If the infor-mation entropy of the overall deformation reached 04770the dammoveddownstreamand in the state of primarywarn-ing If the information entropy of the overall deformationreached minus04773 the dam moved upstream and in the stateof primary warning (Table 11) They all fell into intervalscalculated by fuzzy methodologyThe analysis shows that themethod brought up in this paper is reasonable and scientificAlso the analysis shows the physical meaning of the fuzzywarning indexUnder the action of the unfavorable load com-bination and the influence of the complex random factorsthe maximum entropy andminimum information entropy ofthe overall deformation lie in this interval Considering theinfluences of random factorsmultistage fuzzywarning valueswere receptive and safe
In this paper the overall deformation of 21 buttresswas analyzed through the theoretical method proposedInfluences of random factors on the warning value wereconsidered and multistage fuzzy warning values were deter-mined If the information entropy of overall deformationis in (04763 04775) the dam moved downstream and in
minus2399e minus 003minus1296e minus 003minus1940e minus 0049084e minus 0042011e minus 0033113e minus 0034216e minus 0035318e minus 0036421e minus 0037523e minus 0038625e minus 003
XY
Z
Figure 15The results of finite element calculation ofworking condi-tion 1
XY
Z minus6007e minus 004minus5014e minus 004minus4021e minus 004minus3028e minus 004minus2035e minus 004minus1042e minus 004minus4872e minus 0069443e minus 0051937e minus 0042930e minus 0043923e minus 004
Figure 16 The results of finite element calculation of workingcondition 2
the state of primary warning If the information entropy ofoverall deformation is in (04757 04763) the dam moveddownstream and in the state of secondary warning If theinformation entropy of overall deformation is in (minus04779minus04768) the dam moved upstream and in the state ofprimary warning If the information entropy of the over-all deformation is in (minus04768 minus04763) the dam movedupstream and in the state of secondary warning
7 Conclusion
This paper presented multistage warning indicators of con-crete dam space and considered the influences of complex
Mathematical Problems in Engineering 11
Table 8 Multistage fuzzy warning values of concrete dam
The direction of deformation Confidence levelThe primary warning indicator The secondary warning indicator
Downstream (04763 04775) (04757 04763)Upstream (minus04779 minus04768) (minus04768 minus04763)
Table 9 Material parameter of the dam
Structure Density (kgm) Poissonrsquos ratio Elastic modulus (GPa)Concrete face slab 2400 0167 24Buttress 2400 0167 24Partition wall 2400 0167 24Reinforced concrete block 2400 0160 22Foundation rock mass 2700 0175 12Diorite-dyke 2000 03 115Crushed zone 2000 03 029Horizontal joints 2000 03 07
Table 10 The results of finite element calculation of displacementof observation point (mm)
Observation point Working condition 1 Working condition 2G21 0785 minus059927 0654 minus038628 0356 0114
Table 11 The primary warning indicators of concrete dam
The direction of deformation The primary warning indicatorDownstream 04770Upstream minus04773
random factors The results of the specific studies are asfollows
(1) Influences of fuzziness and randomness of randomfactors on the long-term service of dam were dis-cussed a fuzziness indicator that measures the influ-ence of random factors on monitoring value wasconstructed through cloud model
(2) Equivalent model of overall deformation was pro-posed In the model the overall deformation ofdam was regarded as a deformation system whereeach observation point had different contributions(weights) and affected one another Based on entropytheory a space information entropy that can measurethe overall deformation condition was established
(3) Multistage warning indicators against overall defor-mation of concrete dam under the influences of fuzzi-ness and randomness were determined and nondeter-ministic optimal control of the indicator was achievedto improve the competence of warning against thedeformation of concrete dam
Competing Interests
No conflict of interests exits in the submission of this paper
Acknowledgments
This paper was financially supported by National Natu-ral Science Foundation of China (Grants nos 4132300151139001 51579086 51379068 51279052 51579083 and51209077) Jiangsu Natural Science Foundation (Grants nosBK20140039 and BK2012036) Research Fund for the Doc-toral Program of Higher Education of China (Grant no20130094110010) the Ministry of Water Resources PublicWelfare Industry Research Special Fund Project (Grantsnos 201201038 and 201301061) Jiangsu Province ldquo333 High-Level Personnel Training Projectrdquo (Grants nos BRA2011179and BRA2011145) Project Funded by the Priority AcademicProgram Development of Jiangsu Higher Education Institu-tions (Grant no YS11001) Jiangsu Province ldquo333 High-LevelPersonnel Training Projectrdquo (Grant no 2017-B08037) JiangsuProvince ldquoSix Talent Peaksrdquo Project (Grant no JY-008) andFundamental Research Funds for the Central Universities(Grants nos 2016B04114 and 2015B25414)
References
[1] F BenedettoGGiunta andLMastroeni ldquoAmaximumentropymethod to assess the predictability of financial and commoditypricesrdquo Digital Signal Processing vol 46 pp 19ndash31 2015
[2] S Muraro A Madaschi and A Gajo ldquoOn the reliability of 3Dnumerical analyses on passive piles used for slope stabilisationin frictional soilsrdquo Geotechnique vol 64 no 6 pp 486ndash4922014
[3] W J Yoon and K S Park ldquoA study on the market instabilityindex and risk warning levels in early warning system foreconomic crisisrdquo Digital Signal Processing vol 29 pp 35ndash442014
[4] D Tonini ldquoObserved behavior of several leakier arch damsrdquoJournal of the Power Division vol 82 no 12 pp 135ndash139 1956
12 Mathematical Problems in Engineering
[5] R Salgado and D Kim ldquoReliability analysis of load andresistance factor design of slopesrdquo Journal of Geotechnical andGeoenvironmental Engineering vol 140 no 1 pp 57ndash73 2014
[6] Y C Gu and S J Wang ldquoStudy on the early-warning thresholdof structural instability unexpected accidents of reservoir damrdquoJournal of Hydraulic Engineering vol 40 no 12 pp 1467ndash14722009
[7] H Z Su F Wang and H P Liu ldquoEarly-warning index for damservice behavior based on POT modelrdquo Journal of HydraulicEngineering vol 43 no 8 pp 974ndash986 2012
[8] L Chen Mechanics performance with parameters changing inspace amp monitoring model of roller compacted concrete dam[PhD thesis] Hohai University 2006
[9] A B Li L F Zhang Q L Wang B Li Z Li and Y WangldquoInformation theory in nonlinear error growth dynamics andits application to predictability taking the Lorenz system as anexamplerdquo Science China Earth Sciences vol 56 no 8 pp 1413ndash1421 2013
[10] O Victor ldquoThe theory of cloudsrdquo Library Journal vol 132 no13 p 62 2007
[11] M Yang H Z Su and X Q Yan ldquoComputation and analysis ofhigh rocky slope safety in a water conservancy projectrdquoDiscreteDynamics in Nature and Society vol 2015 Article ID 197579 11pages 2015
[12] C Ricotta and G C Avena ldquoEvaluating the degree of fuzzi-ness of thematic maps with a generalized entropy functiona methodological outlookrdquo International Journal of RemoteSensing vol 23 no 20 pp 4519ndash4523 2002
[13] V I Khvorostyanov and J A Curry ldquoParameterization ofhomogeneous ice nucleation for cloud and climate modelsbased on classical nucleation theoryrdquo Atmospheric Chemistryand Physics vol 12 no 19 pp 9275ndash9302 2012
[14] H Z Su Z P Wen and Z R Wu ldquoStudy on an intelligentinference engine in early-warning system of damhealthrdquoWaterResources Management vol 25 no 6 pp 1545ndash1563 2011
[15] M Kaminski ldquoProbabilistic entropy in homogenization ofthe periodic fiber-reinforced composites with random elasticparametersrdquo International Journal for Numerical Methods inEngineering vol 90 no 8 pp 939ndash954 2012
[16] E Czogała and J Łeski ldquoApplication of entropy and energymeasures of fuzziness to processing of ECG signalrdquo Fuzzy Setsand Systems vol 97 no 1 pp 9ndash18 1998
[17] K Wu and J L Jin ldquoProjection pursuit model for evaluation ofregion water resource security based on changeable weight andinformationrdquo Resources and Environment in the Yangtze Basinvol 20 no 9 pp 1085ndash1091 2011
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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Decision SciencesAdvances in
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Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
Weight of surveypoints 1
Overall deformation of concrete dam
Deformation ofsurvey points 1
Deformation ofsurvey points n
Deformation ofsurvey points 2
Weight of surveypoints 2
Weight of surveypoints n
middot middot middot
middot middot middot
Figure 3 Diagram of overall deformation system of concrete dam
Step 2 The normalized value 119863lowast119894119895
was projected to unitdirection119875 119875 = (119901
1 1199012 119901
119895) and1199012
1+1199012
2+sdot sdot sdot+119901
2
119895= 1The
indicator function of the projection 119866(119894) was constructed
119866 (119894) =
119898
sum
119895=1
119901119895119863lowast
119894119895 (119894 = 1 2 119899) (8)
Step 3 The objective function of the projection was con-structed The best direction for the projection was estimatedby solving the maximization problem of the objective func-tion in the constraint condition
Objective function Max 119867 (119901) = 119878119866sdot 119876119866
Constraint condition119898
sum
119895=1
1199012
119895= 1
(9)
In this equation 119878119866is the divergent degree of the projection
119876119866is the local density of 1D data points along 119875 and is
expressed as follows
119878119866= [sum119899
119894=1(119866 (119894) minus 119892 (119894))
2
119899 minus 1]
05
119876119866=
119899
sum
119894=1
119899
sum
119895=1
(119877 minus 119903119894119895) sdot 119891 (119877 minus 119903
119894119895)
(10)
where119892(119894) is themean value of this sequence119877 is the windowradius of the local density and 119877 = 01119878
119866in this paper 119903
119894119895is
the distance between two projection values and 119891(119905) is theunit step function 119891(119905) is equal to 1 as 119905 is greater than 0Otherwise 119891(119905) is equal to 0
Step 4 The projection value of one sample point was com-puted by substituting the best direction 119875lowast into (8) 120596
119894can be
calculated by normalizing the projection value
120596119894=
119866lowast
(119894)
sum119899
119895=1119866lowast (119895)
119894 = 1 2 119899 (11)
4 Study on Equivalent Model ofDamrsquos Overall Deformation
The overall deformation of a dam can be considered a defor-mation system of feature points with different contributionsthat influence one another as well as observation points ofthe deformation as feature pointsThe deformation conditionwas analyzed systemically and the overall deformation wasexpressed by the evolution of equation of all feature pointsThe deformation condition was described qualitatively bythe tectonic type of information entropy The absolute valueof the information entropy measures the danger level ofthe deformation value Smaller absolute value means higherdanger level The positive and negative values indicate thedirection of the deformation A positive value correspondsto downstream deformation whereas a negative value corre-sponds to upstream deformation The influences of randomfactors were considered and the fuzzy information entropywas constructed
41 Constructing Fuzzy Information Entropy of SingleMeasured Value The downstream deformation is positivewhereas the upstream deformation is negative
When the observation point moves downstream make120583119894119895= int119909119894119895
minusinfin
119891(120589)119889120589 and according to the definition of infor-mation entropy the information entropy of 119909
119894119895is expressed
as (4)Considering the influence of Δ
119894119895 120583119894119895will float in [1205830
119894119895 1205831
119894119895]
the following is then obtained
1205830
119894119895= int
119909119894119895minus119909119894119895Δ 119894119895
minusinfin
119891 (120589) 119889120589
1205831
119894119895= int
119909119894119895+119909119894119895Δ 119894119895
minusinfin
119891 (120589) 119889120589
(12)
When the observation point moves upstream make 120583119894119895=
int+infin
119909119894119895
119891(120589)119889120589 the information entropy of 119909119894119895is defined as
follows119878119894119895= 120583119894119895ln 120583119894119895+ (1 minus 120583
119894119895) ln (1 minus 120583
119894119895) (13)
Mathematical Problems in Engineering 5
0
Dow
nstre
am d
ispla
cem
ent
Ups
tream
di
spla
cem
ent
105
Sij
1205830
1 minus 1205830 120583
Figure 4 Diagram of 119878119894119895changes with the change of 120583
119894119895if 119864119909ge 0
Considering the influence ofΔ119894119895 120583119894119895floats in [1205830
119894119895 1205831
119894119895]The
following is then obtained
1205830
119894119895= int
+infin
119909119894119895minus119909119894119895Δ 119894119895
119891 (120589) 119889120589
1205831
119894119895= int
+infin
119909119894119895+119909119894119895Δ 119894119895
119891 (120589) 119889120589
(14)
Considering the influences of random factors the infor-mation entropy of 119909
119894119895minus119878119894119895floats in [1198780
119894119895 1198781
119894119895] whichwas defined
as the fuzzy information entropy of 119909119894119895
Take the downstream as an example Influenced by deter-mined and random factors the fuzzy entropy of 119878
119894119895changes
into the range of 119878119894119895in the range of [1205830
119894119895 1205831
119894119895]
42 Methods for Determining the Range of the InformationEntropy of Single Measured Value For Figure 4 the expecta-tion of one deformation monitoring sequence sample at oneobservation point (119864
119909ge 0 119878
119894119895) changes with the change of
120583119894119895 as shown in Figure 4 where 120583
0= int0
minusinfin
119891(120589)119889120589When the dam deforms downstream the change law of
119878119894119895is as follows 119878
119894119895will increase with increasing 120583
119894119895when 120583
119894119895
is in the range of (1205830 05) 119878
119894119895will decrease with decreasing
120583119894119895when 120583
119894119895is in the range of (05 1) when 120583
119894119895= 05 119878
119894119895will
reach themaximumvalue If 05 is in the range of [1205830119894119895 1205831
119894119895] the
maximum of 119878119894119895is 1198781119894119895when 120583
119894119895= 05 and its minimum value
is 1198780119894119895at the endpoint If 05 is not in the range of [1205830
119894119895 1205831
119894119895]
119878119894119895will have its maximum value 1198781
119894119895and minimum value 1198780
119894119895
at endpoints When the dam deforms upstream 119878119894119895will rise
with the rise of 120583119894119895and it will have its maximum value 1198781
119894119895and
minimum value 1198780119894119895at endpoints
The expectation of one deformationmonitoring sequencesample at one observation point (119864
119909lt 0 119878
119894119895) changes with
01
05
Dow
nstre
amU
pstre
am d
ispla
cem
ent
Sij
1205830
1 minus 1205830 120583
disp
lace
men
t
Figure 5 Diagram of 119878119894119895changes with the change of 120583
119894119895if 119864119909lt 0
the change of 120583119894119895as shown in Figure 5 where 120583
0= int0
minusinfin
119891(120589)119889120589
and Figure 5 is presented for 119878119894119895changes with the change of
120583119894119895When the damdeforms downstream the change law of 119878
119894119895
is as follows 119878119894119895will increase with decreasing 120583
119894119895and will have
its maximum value 1198781119894119895and minimum value 1198780
119894119895at endpoints
When the damdeforms upstream the figure shows that when120583119894119895is in the range of (1minus120583
0 05) 119878
119894119895will increase with the drop
of 120583119894119895 when 120583
119894119895is in the range of (05 1) 119878
119894119895will decrease with
the increase of 120583119894119895 when 120583
119894119895= 05 119878
119894119895will reach theminimum
value If 05 is contained in the range of [1205830119894119895 1205831
119894119895] 119878119894119895will have
its minimum value 1198780119894119895at 120583119894119895= 05 and have its maximum
value 1198781119894119895at endpoint if 05 is not contained in the range 119878
119894119895
will have its maximum value 1198781119894119895and minimum value 1198780
119894119895at
endpoints
43 Construction of the Fuzzy Information Entropy of OverallDeformation On the basis of the above results the expres-sion of information entropy of overall deformation can bededucedThe contribution of the order degree of observationpoint 119894 is 120596
119894120583119894119895 make 1205831
119894119895= 120583119894119895and 1205832
119894119895= 1 minus 120583
119894119895 According to
the broad definition of information entropy when the dammove deforms downstream the expression of informationentropy of overall deformation is expressed as follows
119878119895= minus
119898
sum
119894=1
2
sum
119896=1
120596119894120583119896
119894119895ln (120596119894120583119896
119894119895)
= minus
119899
sum
119894=1
2
sum
119896=1
120596119894120583119896
119894119895(ln120596119894+ ln 120583119896
119894119895)
= minus
119899
sum
119894=1
2
sum
119896=1
120596119894120583119896
119894119895ln120596119894minus
119899
sum
119894=1
2
sum
119896=1
120596119894120583119896
119894119895ln 120583119896119894119895
6 Mathematical Problems in Engineering
Choice of sample
Input sample eigenvalues
Calling cloud model program
Generating range of cloud fall
Calculating the fuzzinessof all measured value
Constructing deformed entropyof one measured value
Calculating weight of singleobservation point
Constructing fuzzy deformedentropy of overall deformation
Constructing fuzzy deformedentropy of one measured value
Ex EnHe
Figure 6 Computational process of fuzzy information entropy of overall deformation
= minus
119899
sum
119894=1
120596119894ln120596119894
2
sum
119896=1
120583119896
119894119895minus
119899
sum
119894=1
120596119894
2
sum
119896=1
120583119896
119894119895ln 120583119896119894119895
= minus
119899
sum
119894=1
120596119894ln120596119894+
119899
sum
119894=1
120596119894119878119894119895
(15)
When the dam move deforms upstream the expressionof the information entropy of overall deformation is
119878119895=
119898
sum
119894=1
2
sum
119896=1
120596119894120583119896
119894119895ln (120596119894120583119896
119894119895) =
119899
sum
119894=1
120596119894ln120596119894+
119899
sum
119894=1
120596119894119878119894119895 (16)
Therefore the expression of information entropy of over-all deformation is defined as follows
119878119895=
minus
119899
sum
119894=1
120596119894ln120596119894+
119899
sum
119894=1
120596119894119878119894119895 119878119894119895ge 0
119899
sum
119894=1
120596119894ln120596119894+
119899
sum
119894=1
120596119894119878119894119895 119878
119894119895lt 0
(17)
The absolute value of the information entropy of the over-all deformationmeasures the danger level of the deformationthat is a smaller absolute value means a higher danger levelpositive and negative values stand for the direction of thedeformation a positive value means downstream deforma-tion whereas a negative value means upstream deformation
Considering the influences of random factors the fuzzyinformation entropy of 119909
119894119895is [1198780119894119895 1198781
119894119895] the fuzzy information
entropy of overall deformation can be illustrated through(17) Computational process of fuzzy information entropy ofoverall deformation is shown in Figure 6
5 Proposed Multistage FuzzyInformation Entropy of OverallDeformation Warning Indicators
Horizontal displacement of dam crest changes in an annualcycle ldquoupstream and downstream switchrdquo Therefore thisdisplacement should be in a certain scale and be controlledunder somemonitoring indicators for the safe damoperation
In the case of downstream displacement the primaryfuzzy warning indicator 1205751015840
1is defined as 1205751015840
1= (1205750
1 1205751
1) 12057501is
the lower limit of this indicator and 12057511is the upper limit
the secondary indicator 12057510158402is 12057510158402= (1205750
2 1205751
2) where 1205750
2is
the lower limit of this indicator and 12057512is the upper limit
When 12057511gt 1205750
2 a cross phenomenon appears in the primary
indicator and secondary indicator when both of them shouldbe categorized according to the membership of displacementmeasured 120575lowast was introduced because the membership ofdisplacement at this point is the same Figure 7 showsdiagram of multistage fuzzy information entropy warningindicators
The primary fuzzy warning indicator is 12057510158401= (1205750
1 120575lowast
) andthe secondary is 1205751015840
2= (120575lowast
1205751
2)
If the deformation value is in (12057501 1205751
1) or (1205750
1 120575lowast
) the damis in the state of primary warning if the value is in (1205750
2 1205751
2) or
(1205750
2 120575lowast
) the dam is in the state of secondary warningThe time sequence of deformation at each observation
point was analyzed by using the above theoretical methodThe lower and upper limits of the fuzzy information entropyof overall deformation affected by Δ
119894119895will be 1198780
119895 and 1198781
119895
Considering the damrsquos long-term service when the dammoves downstream the lower limit 1198780
119898119895 and upper limit
Mathematical Problems in Engineering 7
1Degree of membership
Displacement
1Degree of membership
Displacement
12057512120575021205751112057501
1205751212057502 1205751112057501 120575lowast
Figure 7Diagramofmultistage fuzzy information entropywarningindicators
1198781
119898119895 were selected and when the dam moves upstream
the lower limit 1198770119898119895 and upper limit 1198771
119898119895 were selected
1198780
119898119895 1198781119898119895 1198770119898119895 and 1198771
119898119895 are random variables and four
subsample spaces with the sample size of 119873 can be obtainedby the following
1198780
= 1198780
1198981 1198780
1198982 119878
0
119898119898
1198781
= 1198781
1198981 1198781
1198982 119878
1
119898119898
1198770
= 1198770
1198981 1198770
1198982 119877
0
119898119898
1198771
= 1198771
1198981 1198771
1198982 119877
1
119898119898
(18)
Shapiro-Wilk test andKolmogorov-Smirnov test can bothtest whether the samples obey normal distribution or not Butthe Kolmogorov-Smirnov test is applicable to fewer samplesIt can not only test if the samples are subject to normaldistribution but also test if samples are subject to otherdistributions The basic idea of the K-S test is to comparethe cumulative frequency of the observed value (119865
119899(119909)) with
the assumed theoretical probability distribution (119865119909(119909)) to
construct statisticsAccording to the method of empirical distribution func-
tion segmented cumulative frequency is obtained by usingthe following formula
119865119899(119909) =
0 119909 lt 119909119894
119894
119899 119909119894le 119909 lt 119909
119894+1
1 119909 ge 119909
(19)
In the formula 1199091 1199092 119909
119899is sample data after arrange-
ment The sample size is 119899
In the full range of random variable 119883 the maximumdifference between 119865
119899(119909) and 119865
119909(119909) is
119863119899= max 1003816100381610038161003816119865119909 (119909) minus 119865119899 (119909)
1003816100381610038161003816 lt 119863120573
119899 (20)
In the formula 119863119899is a random variable whose distribu-
tion depends on 119899119863120573119899is critical value for a significant level 120573
It is considered that the distribution to be used at a significantlevel 120573 cannot be resisted otherwise it should be rejected
The distribution formwas tested through the K-Smethodto determine the probability density function Fuzzy warningindicator was then determinedwith different significant levelIn dam safety evaluation significant level 120572 is the probabilityof the dam failure Supposing 119878
119898is the extreme of the
information entropy of the upstream overall deformation if119878 gt 119878
119898 the probability of the dam failure is 119875(119878 gt 119878
119898) =
120572 = intinfin
119878119898
119891(119909)119889119909 and the reliability index of dam failure is1 minus 120572 According to the dam importance different failureprobability is set and the multistage warning indicators wereidentified
6 Example Analysis
61 Project Profile One flat-slab deck dam built with rein-forced concrete is an important part of one river basin cascadeexploitation The elevation of this dam crest is 13770m andthe height of biggest part is about 43m the crest runs 2250min length and is made of 27 flat-slab buttresses with a span of75m The space between the left side of 2 buttress and theright side of 9 buttress is the joint part the overflow buttressis located from the 9 buttress to the 20 buttress the rest isthe water-retaining buttressTheworkshop buttress is locatedfrom the 5 buttress to the 8 buttress In this dam the levelof deadwater is 1220m the normal highwater level is 1310m(in practice it is 1290m) the design flood level is 1367m andthe check flood level is 1375m To monitor the displacementof this dam a direct plumb line and an inverted plumb linewere arranged in four buttresses 4 9 21 and 24There isa crushed zone under the dam foundation where occurrenceis N20∘sim25∘W SWang70∘sim80∘ the maximum width is about3m and the narrowest place is about 1mThere is an elevationclip joint mud at level 91m
The deformation field characterized by the observationpoint at 21 should be typical and is a key point because it isin the riverbedThus the observation point G21 at the heightof 134m along the direct plumb line at 21 was analyzed aswell as point 27 at the height of 118m along this line andpoint 28 at the height of 107mAll these valuesmeasuredweretransformed into absolute displacement The arrangement ofeach point is shown in Figure 8 The daily monitoring dataseries is from January 1 2003 to December 31 2013
In this paper two-stage warning indicators were setaccording to the practical running of this project and danger120572 = 5 is the primary warning that is mainly used to discri-minate and handle early dangerous case and the reliabilityindex of dam failure is 95 whereas 120572 = 1 is the secondarywarning that is mainly used to determine grave danger andprevent urgent danger and the reliability index of dam failureis 99
8 Mathematical Problems in Engineering
G21 G9 G4G24
25
30
2631
28
27
3233
Figure 8 The arrangement of observation point
Ups
tream
wat
er le
vel
623
198
4
121
419
89
66
1995
112
620
00
519
200
6
119
201
1
11
1979
Date
120124128132
Figure 9 The process line of upstream water level
11
2007
11
2008
11
2006
123
120
08
11
2005
123
120
09
123
120
10
123
120
11
123
020
12
123
020
13
Data
05
101520253035
Tem
pera
ture
Figure 10 The process line of temperature
62 Calculating the Contribution of Deformation at theObservation Point to the Overall Deformation Figures 9-10 show the upstream stage hydrograph and temperaturestage hydrograph respectively The water stage remainedunchanged whereas the temperature changed in the annualcycle Figure 11 shows the relationship between informationentropy of overall deformation and temperature in 2007 Anegative correlation exists between the overall deformationof 21 buttress and temperature that is an increase in tem-perature corresponds to the decrease in upstream or down-stream displacement and a decrease in temperature meansan increase in the upstream or downstream displacementFigure 12 shows the correlation between the displacementat observation point G21 and the temperature The overalldeformation law is roughly identical with that at observationpoint G21
Figure 12 reveals that the temperature obviously influ-enced overall deformation that is the temperature canchange the contribution of single observation point to theoverall deformation Temperature change was divided intothe stage of temperature rise and stage of temperature drop
2011
11
2011
21
2011
31
2011
41
2011
51
2011
61
2011
71
2011
81
2011
91
2011
10
1
2011
11
1
2011
12
1
TemperatureDeformed entropy
0070
140210280
Tem
pera
ture
minus080minus076minus072minus068minus064minus060
Def
orm
ed en
tropy
Figure 11 The relationship between information entropy of overalldeformation and temperature in 2011
2011
11
2011
13
1
2011
32
2011
41
2011
51
2011
53
1
2011
63
0
2011
73
0
2011
82
9
2011
92
8
2011
10
28
2011
11
27
2011
12
27
Horizontal displacementTemperature
05
101520253035
Tem
pera
ture
minus16minus12minus08minus04004
Hor
izon
tal
disp
lace
men
t
Figure 12 The relationship between horizontal displacement andtemperature of observation point G21 in 2011
The weight of deformation at observation point in two stageswas calculated The results are shown in the Table 1
63 Results of Information Entropy of Overall DeformationRange of cloud fall of the displacement at G21 27 and28 is shown in Figure 13 The boundary values of mostdangerous information entropy from 2003 to 2013 are shownin Tables 2ndash5The absolute value means the degree of dangerdownstream is set as negative value In the significance level120572 = 005 7 kinds of common distribution (lognormal distri-bution normal distribution uniform distribution triangulardistribution exponential distribution 120574 distribution and 120573distribution) were used to carry on hypothesis test for 1198780
119898119895
1198781
119898119895 1198770119898119895 and 1198771
119898119895 with K-S method The maximum 119863
119899
of each sequencewas obtained and comparedwith the criticalvalue 119863005
119899of the significant level 005 to determine the type
Mathematical Problems in Engineering 9
Table 1 The weight table of observation point
Observation point Altitude The period of temperature rise The period of temperature dropG21 134m 0357 039227 118m 0331 031628 107m 0312 0292
Table 2 The lower limit of the most dangerous information entropy of downstream overall deformation
Year 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013Information entropy 04784 04837 04767 04784 04969 04887 04969 05102 05082 04799 04739
Table 3 The upper limit of the most dangerous information entropy of downstream overall deformation
Year 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013Information entropy 04791 04845 04770 04786 04971 04899 04974 05186 05161 04817 04769
Table 4 The lower limit of the most dangerous information entropy of upstream overall deformation
Year 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013Information entropy minus05433 minus05491 minus04773 minus05083 minus05067 minus04784 minus05080 minus04839 minus04836 minus04825 minus05020
Table 5 The lower limit of the most dangerous information entropy of upstream overall deformation
Year 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013Information entropy minus05339 minus05440 minus04770 minus05079 minus05055 minus04770 minus05040 minus04822 minus04806 minus04754 minus04766
Cloud droplets
minus15
minus14
minus13
minus12
minus11
minus1
minus09
minus08
minus07
minus06
minus05
minus04
minus03
minus02
minus01
010
02
03
04
05
06
07
08
09 1
00102030405060708091
Cer
tain
ty
G21 27 28
Figure 13 Range of cloud fall of the displacement at G21 27 and 28
of the best distribution K-S test results are shown in Table 6Multistage fuzzy warning values are presented in Table 7
K-S test shows that 1198780119898119895 1198781119898119895 1198770119898119895 and 1198771
119898119895 satisfy
normal distributionThe probability density function of the sequence is
119891 (119909) =1
radic21205871205902exp(minus
(119909 minus 120583)2
120590) (21)
Parameter values of (21) are presented in Table 7In downstream deformation if 120572 = 5 the primary
warning indicator is (04763 04775) if 120572 = 1 thesecondary warning indicator is (04757 04763) 120575lowast is used as
the boundary value when two indicators overlap In the caseof upstream deformation if 120572 = 5 the primary warningindicator is (minus04779 minus04768) if 120572 = 1 the secondarywarning indicator is (minus04768 minus04763) (Table 8)
64 The Structure Calculation of Monitoring Index of DamHorizontal Displacement According to the actual situationthree-dimensional finite element model of the dam is estab-lished According to the structure and basic geologicalconditions of 21 dam section the scope of finite elementcalculation model can be got as follows taking 15 times ashigh dam in the upstream direction taking 15 times as highdam in the upstream direction and taking 1 time as highdam below the dam foundation The model is constitutedof 11445 nodes and 8571 units The unit type is 6 sides 8nodes isoparametric element The deformation observationdata analysis shows the dam under the condition of lowtemperature and highwater level there is larger displacementin the downstream when in high temperature and in lowwater level there is larger displacement in the upstream(Table 9 Figure 14) In view of the actual situation of thedam observation and data analysis results the load conditionselection is as follows (Table 10 Figures 15 and 16)
Working condition 1 normal water level 1290m andmaximum temperature dropWorking condition 2 dead water level 1220m andmaximum temperature rise
10 Mathematical Problems in Engineering
Table 6 K-S test results
Probability distribution 1198780
1198981198951198781
1198981198951198770
1198981198951198771
119898119895
Lognormal distribution 026 011 023 028Normal distribution 011 008 026 022Uniform distribution 074 155 088 068Triangular distribution 053 054 059 061Exponential distribution 041 042 055 035120574 distribution 037 036 031 033120573 distribution 068 073 086 063119863005
119899034 034 029 029
The most reasonable probability distribution Normal distribution Normal distribution Normal distribution Normal distribution
Table 7 Parameter values of the probability density function
Data series Parameter values120583 120590
2
1198780
119898119895 0488355 00129
1198781
119898119895 0490627 00151
1198770
119898119895 minus050210 00249
1198771
119898119895 minus049674 00245
Figure 14 Finite element model of the dam
The primary warning indicators of concrete dam wereobtained by calculation methods for structures If the infor-mation entropy of the overall deformation reached 04770the dammoveddownstreamand in the state of primarywarn-ing If the information entropy of the overall deformationreached minus04773 the dam moved upstream and in the stateof primary warning (Table 11) They all fell into intervalscalculated by fuzzy methodologyThe analysis shows that themethod brought up in this paper is reasonable and scientificAlso the analysis shows the physical meaning of the fuzzywarning indexUnder the action of the unfavorable load com-bination and the influence of the complex random factorsthe maximum entropy andminimum information entropy ofthe overall deformation lie in this interval Considering theinfluences of random factorsmultistage fuzzywarning valueswere receptive and safe
In this paper the overall deformation of 21 buttresswas analyzed through the theoretical method proposedInfluences of random factors on the warning value wereconsidered and multistage fuzzy warning values were deter-mined If the information entropy of overall deformationis in (04763 04775) the dam moved downstream and in
minus2399e minus 003minus1296e minus 003minus1940e minus 0049084e minus 0042011e minus 0033113e minus 0034216e minus 0035318e minus 0036421e minus 0037523e minus 0038625e minus 003
XY
Z
Figure 15The results of finite element calculation ofworking condi-tion 1
XY
Z minus6007e minus 004minus5014e minus 004minus4021e minus 004minus3028e minus 004minus2035e minus 004minus1042e minus 004minus4872e minus 0069443e minus 0051937e minus 0042930e minus 0043923e minus 004
Figure 16 The results of finite element calculation of workingcondition 2
the state of primary warning If the information entropy ofoverall deformation is in (04757 04763) the dam moveddownstream and in the state of secondary warning If theinformation entropy of overall deformation is in (minus04779minus04768) the dam moved upstream and in the state ofprimary warning If the information entropy of the over-all deformation is in (minus04768 minus04763) the dam movedupstream and in the state of secondary warning
7 Conclusion
This paper presented multistage warning indicators of con-crete dam space and considered the influences of complex
Mathematical Problems in Engineering 11
Table 8 Multistage fuzzy warning values of concrete dam
The direction of deformation Confidence levelThe primary warning indicator The secondary warning indicator
Downstream (04763 04775) (04757 04763)Upstream (minus04779 minus04768) (minus04768 minus04763)
Table 9 Material parameter of the dam
Structure Density (kgm) Poissonrsquos ratio Elastic modulus (GPa)Concrete face slab 2400 0167 24Buttress 2400 0167 24Partition wall 2400 0167 24Reinforced concrete block 2400 0160 22Foundation rock mass 2700 0175 12Diorite-dyke 2000 03 115Crushed zone 2000 03 029Horizontal joints 2000 03 07
Table 10 The results of finite element calculation of displacementof observation point (mm)
Observation point Working condition 1 Working condition 2G21 0785 minus059927 0654 minus038628 0356 0114
Table 11 The primary warning indicators of concrete dam
The direction of deformation The primary warning indicatorDownstream 04770Upstream minus04773
random factors The results of the specific studies are asfollows
(1) Influences of fuzziness and randomness of randomfactors on the long-term service of dam were dis-cussed a fuzziness indicator that measures the influ-ence of random factors on monitoring value wasconstructed through cloud model
(2) Equivalent model of overall deformation was pro-posed In the model the overall deformation ofdam was regarded as a deformation system whereeach observation point had different contributions(weights) and affected one another Based on entropytheory a space information entropy that can measurethe overall deformation condition was established
(3) Multistage warning indicators against overall defor-mation of concrete dam under the influences of fuzzi-ness and randomness were determined and nondeter-ministic optimal control of the indicator was achievedto improve the competence of warning against thedeformation of concrete dam
Competing Interests
No conflict of interests exits in the submission of this paper
Acknowledgments
This paper was financially supported by National Natu-ral Science Foundation of China (Grants nos 4132300151139001 51579086 51379068 51279052 51579083 and51209077) Jiangsu Natural Science Foundation (Grants nosBK20140039 and BK2012036) Research Fund for the Doc-toral Program of Higher Education of China (Grant no20130094110010) the Ministry of Water Resources PublicWelfare Industry Research Special Fund Project (Grantsnos 201201038 and 201301061) Jiangsu Province ldquo333 High-Level Personnel Training Projectrdquo (Grants nos BRA2011179and BRA2011145) Project Funded by the Priority AcademicProgram Development of Jiangsu Higher Education Institu-tions (Grant no YS11001) Jiangsu Province ldquo333 High-LevelPersonnel Training Projectrdquo (Grant no 2017-B08037) JiangsuProvince ldquoSix Talent Peaksrdquo Project (Grant no JY-008) andFundamental Research Funds for the Central Universities(Grants nos 2016B04114 and 2015B25414)
References
[1] F BenedettoGGiunta andLMastroeni ldquoAmaximumentropymethod to assess the predictability of financial and commoditypricesrdquo Digital Signal Processing vol 46 pp 19ndash31 2015
[2] S Muraro A Madaschi and A Gajo ldquoOn the reliability of 3Dnumerical analyses on passive piles used for slope stabilisationin frictional soilsrdquo Geotechnique vol 64 no 6 pp 486ndash4922014
[3] W J Yoon and K S Park ldquoA study on the market instabilityindex and risk warning levels in early warning system foreconomic crisisrdquo Digital Signal Processing vol 29 pp 35ndash442014
[4] D Tonini ldquoObserved behavior of several leakier arch damsrdquoJournal of the Power Division vol 82 no 12 pp 135ndash139 1956
12 Mathematical Problems in Engineering
[5] R Salgado and D Kim ldquoReliability analysis of load andresistance factor design of slopesrdquo Journal of Geotechnical andGeoenvironmental Engineering vol 140 no 1 pp 57ndash73 2014
[6] Y C Gu and S J Wang ldquoStudy on the early-warning thresholdof structural instability unexpected accidents of reservoir damrdquoJournal of Hydraulic Engineering vol 40 no 12 pp 1467ndash14722009
[7] H Z Su F Wang and H P Liu ldquoEarly-warning index for damservice behavior based on POT modelrdquo Journal of HydraulicEngineering vol 43 no 8 pp 974ndash986 2012
[8] L Chen Mechanics performance with parameters changing inspace amp monitoring model of roller compacted concrete dam[PhD thesis] Hohai University 2006
[9] A B Li L F Zhang Q L Wang B Li Z Li and Y WangldquoInformation theory in nonlinear error growth dynamics andits application to predictability taking the Lorenz system as anexamplerdquo Science China Earth Sciences vol 56 no 8 pp 1413ndash1421 2013
[10] O Victor ldquoThe theory of cloudsrdquo Library Journal vol 132 no13 p 62 2007
[11] M Yang H Z Su and X Q Yan ldquoComputation and analysis ofhigh rocky slope safety in a water conservancy projectrdquoDiscreteDynamics in Nature and Society vol 2015 Article ID 197579 11pages 2015
[12] C Ricotta and G C Avena ldquoEvaluating the degree of fuzzi-ness of thematic maps with a generalized entropy functiona methodological outlookrdquo International Journal of RemoteSensing vol 23 no 20 pp 4519ndash4523 2002
[13] V I Khvorostyanov and J A Curry ldquoParameterization ofhomogeneous ice nucleation for cloud and climate modelsbased on classical nucleation theoryrdquo Atmospheric Chemistryand Physics vol 12 no 19 pp 9275ndash9302 2012
[14] H Z Su Z P Wen and Z R Wu ldquoStudy on an intelligentinference engine in early-warning system of damhealthrdquoWaterResources Management vol 25 no 6 pp 1545ndash1563 2011
[15] M Kaminski ldquoProbabilistic entropy in homogenization ofthe periodic fiber-reinforced composites with random elasticparametersrdquo International Journal for Numerical Methods inEngineering vol 90 no 8 pp 939ndash954 2012
[16] E Czogała and J Łeski ldquoApplication of entropy and energymeasures of fuzziness to processing of ECG signalrdquo Fuzzy Setsand Systems vol 97 no 1 pp 9ndash18 1998
[17] K Wu and J L Jin ldquoProjection pursuit model for evaluation ofregion water resource security based on changeable weight andinformationrdquo Resources and Environment in the Yangtze Basinvol 20 no 9 pp 1085ndash1091 2011
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
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Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
0
Dow
nstre
am d
ispla
cem
ent
Ups
tream
di
spla
cem
ent
105
Sij
1205830
1 minus 1205830 120583
Figure 4 Diagram of 119878119894119895changes with the change of 120583
119894119895if 119864119909ge 0
Considering the influence ofΔ119894119895 120583119894119895floats in [1205830
119894119895 1205831
119894119895]The
following is then obtained
1205830
119894119895= int
+infin
119909119894119895minus119909119894119895Δ 119894119895
119891 (120589) 119889120589
1205831
119894119895= int
+infin
119909119894119895+119909119894119895Δ 119894119895
119891 (120589) 119889120589
(14)
Considering the influences of random factors the infor-mation entropy of 119909
119894119895minus119878119894119895floats in [1198780
119894119895 1198781
119894119895] whichwas defined
as the fuzzy information entropy of 119909119894119895
Take the downstream as an example Influenced by deter-mined and random factors the fuzzy entropy of 119878
119894119895changes
into the range of 119878119894119895in the range of [1205830
119894119895 1205831
119894119895]
42 Methods for Determining the Range of the InformationEntropy of Single Measured Value For Figure 4 the expecta-tion of one deformation monitoring sequence sample at oneobservation point (119864
119909ge 0 119878
119894119895) changes with the change of
120583119894119895 as shown in Figure 4 where 120583
0= int0
minusinfin
119891(120589)119889120589When the dam deforms downstream the change law of
119878119894119895is as follows 119878
119894119895will increase with increasing 120583
119894119895when 120583
119894119895
is in the range of (1205830 05) 119878
119894119895will decrease with decreasing
120583119894119895when 120583
119894119895is in the range of (05 1) when 120583
119894119895= 05 119878
119894119895will
reach themaximumvalue If 05 is in the range of [1205830119894119895 1205831
119894119895] the
maximum of 119878119894119895is 1198781119894119895when 120583
119894119895= 05 and its minimum value
is 1198780119894119895at the endpoint If 05 is not in the range of [1205830
119894119895 1205831
119894119895]
119878119894119895will have its maximum value 1198781
119894119895and minimum value 1198780
119894119895
at endpoints When the dam deforms upstream 119878119894119895will rise
with the rise of 120583119894119895and it will have its maximum value 1198781
119894119895and
minimum value 1198780119894119895at endpoints
The expectation of one deformationmonitoring sequencesample at one observation point (119864
119909lt 0 119878
119894119895) changes with
01
05
Dow
nstre
amU
pstre
am d
ispla
cem
ent
Sij
1205830
1 minus 1205830 120583
disp
lace
men
t
Figure 5 Diagram of 119878119894119895changes with the change of 120583
119894119895if 119864119909lt 0
the change of 120583119894119895as shown in Figure 5 where 120583
0= int0
minusinfin
119891(120589)119889120589
and Figure 5 is presented for 119878119894119895changes with the change of
120583119894119895When the damdeforms downstream the change law of 119878
119894119895
is as follows 119878119894119895will increase with decreasing 120583
119894119895and will have
its maximum value 1198781119894119895and minimum value 1198780
119894119895at endpoints
When the damdeforms upstream the figure shows that when120583119894119895is in the range of (1minus120583
0 05) 119878
119894119895will increase with the drop
of 120583119894119895 when 120583
119894119895is in the range of (05 1) 119878
119894119895will decrease with
the increase of 120583119894119895 when 120583
119894119895= 05 119878
119894119895will reach theminimum
value If 05 is contained in the range of [1205830119894119895 1205831
119894119895] 119878119894119895will have
its minimum value 1198780119894119895at 120583119894119895= 05 and have its maximum
value 1198781119894119895at endpoint if 05 is not contained in the range 119878
119894119895
will have its maximum value 1198781119894119895and minimum value 1198780
119894119895at
endpoints
43 Construction of the Fuzzy Information Entropy of OverallDeformation On the basis of the above results the expres-sion of information entropy of overall deformation can bededucedThe contribution of the order degree of observationpoint 119894 is 120596
119894120583119894119895 make 1205831
119894119895= 120583119894119895and 1205832
119894119895= 1 minus 120583
119894119895 According to
the broad definition of information entropy when the dammove deforms downstream the expression of informationentropy of overall deformation is expressed as follows
119878119895= minus
119898
sum
119894=1
2
sum
119896=1
120596119894120583119896
119894119895ln (120596119894120583119896
119894119895)
= minus
119899
sum
119894=1
2
sum
119896=1
120596119894120583119896
119894119895(ln120596119894+ ln 120583119896
119894119895)
= minus
119899
sum
119894=1
2
sum
119896=1
120596119894120583119896
119894119895ln120596119894minus
119899
sum
119894=1
2
sum
119896=1
120596119894120583119896
119894119895ln 120583119896119894119895
6 Mathematical Problems in Engineering
Choice of sample
Input sample eigenvalues
Calling cloud model program
Generating range of cloud fall
Calculating the fuzzinessof all measured value
Constructing deformed entropyof one measured value
Calculating weight of singleobservation point
Constructing fuzzy deformedentropy of overall deformation
Constructing fuzzy deformedentropy of one measured value
Ex EnHe
Figure 6 Computational process of fuzzy information entropy of overall deformation
= minus
119899
sum
119894=1
120596119894ln120596119894
2
sum
119896=1
120583119896
119894119895minus
119899
sum
119894=1
120596119894
2
sum
119896=1
120583119896
119894119895ln 120583119896119894119895
= minus
119899
sum
119894=1
120596119894ln120596119894+
119899
sum
119894=1
120596119894119878119894119895
(15)
When the dam move deforms upstream the expressionof the information entropy of overall deformation is
119878119895=
119898
sum
119894=1
2
sum
119896=1
120596119894120583119896
119894119895ln (120596119894120583119896
119894119895) =
119899
sum
119894=1
120596119894ln120596119894+
119899
sum
119894=1
120596119894119878119894119895 (16)
Therefore the expression of information entropy of over-all deformation is defined as follows
119878119895=
minus
119899
sum
119894=1
120596119894ln120596119894+
119899
sum
119894=1
120596119894119878119894119895 119878119894119895ge 0
119899
sum
119894=1
120596119894ln120596119894+
119899
sum
119894=1
120596119894119878119894119895 119878
119894119895lt 0
(17)
The absolute value of the information entropy of the over-all deformationmeasures the danger level of the deformationthat is a smaller absolute value means a higher danger levelpositive and negative values stand for the direction of thedeformation a positive value means downstream deforma-tion whereas a negative value means upstream deformation
Considering the influences of random factors the fuzzyinformation entropy of 119909
119894119895is [1198780119894119895 1198781
119894119895] the fuzzy information
entropy of overall deformation can be illustrated through(17) Computational process of fuzzy information entropy ofoverall deformation is shown in Figure 6
5 Proposed Multistage FuzzyInformation Entropy of OverallDeformation Warning Indicators
Horizontal displacement of dam crest changes in an annualcycle ldquoupstream and downstream switchrdquo Therefore thisdisplacement should be in a certain scale and be controlledunder somemonitoring indicators for the safe damoperation
In the case of downstream displacement the primaryfuzzy warning indicator 1205751015840
1is defined as 1205751015840
1= (1205750
1 1205751
1) 12057501is
the lower limit of this indicator and 12057511is the upper limit
the secondary indicator 12057510158402is 12057510158402= (1205750
2 1205751
2) where 1205750
2is
the lower limit of this indicator and 12057512is the upper limit
When 12057511gt 1205750
2 a cross phenomenon appears in the primary
indicator and secondary indicator when both of them shouldbe categorized according to the membership of displacementmeasured 120575lowast was introduced because the membership ofdisplacement at this point is the same Figure 7 showsdiagram of multistage fuzzy information entropy warningindicators
The primary fuzzy warning indicator is 12057510158401= (1205750
1 120575lowast
) andthe secondary is 1205751015840
2= (120575lowast
1205751
2)
If the deformation value is in (12057501 1205751
1) or (1205750
1 120575lowast
) the damis in the state of primary warning if the value is in (1205750
2 1205751
2) or
(1205750
2 120575lowast
) the dam is in the state of secondary warningThe time sequence of deformation at each observation
point was analyzed by using the above theoretical methodThe lower and upper limits of the fuzzy information entropyof overall deformation affected by Δ
119894119895will be 1198780
119895 and 1198781
119895
Considering the damrsquos long-term service when the dammoves downstream the lower limit 1198780
119898119895 and upper limit
Mathematical Problems in Engineering 7
1Degree of membership
Displacement
1Degree of membership
Displacement
12057512120575021205751112057501
1205751212057502 1205751112057501 120575lowast
Figure 7Diagramofmultistage fuzzy information entropywarningindicators
1198781
119898119895 were selected and when the dam moves upstream
the lower limit 1198770119898119895 and upper limit 1198771
119898119895 were selected
1198780
119898119895 1198781119898119895 1198770119898119895 and 1198771
119898119895 are random variables and four
subsample spaces with the sample size of 119873 can be obtainedby the following
1198780
= 1198780
1198981 1198780
1198982 119878
0
119898119898
1198781
= 1198781
1198981 1198781
1198982 119878
1
119898119898
1198770
= 1198770
1198981 1198770
1198982 119877
0
119898119898
1198771
= 1198771
1198981 1198771
1198982 119877
1
119898119898
(18)
Shapiro-Wilk test andKolmogorov-Smirnov test can bothtest whether the samples obey normal distribution or not Butthe Kolmogorov-Smirnov test is applicable to fewer samplesIt can not only test if the samples are subject to normaldistribution but also test if samples are subject to otherdistributions The basic idea of the K-S test is to comparethe cumulative frequency of the observed value (119865
119899(119909)) with
the assumed theoretical probability distribution (119865119909(119909)) to
construct statisticsAccording to the method of empirical distribution func-
tion segmented cumulative frequency is obtained by usingthe following formula
119865119899(119909) =
0 119909 lt 119909119894
119894
119899 119909119894le 119909 lt 119909
119894+1
1 119909 ge 119909
(19)
In the formula 1199091 1199092 119909
119899is sample data after arrange-
ment The sample size is 119899
In the full range of random variable 119883 the maximumdifference between 119865
119899(119909) and 119865
119909(119909) is
119863119899= max 1003816100381610038161003816119865119909 (119909) minus 119865119899 (119909)
1003816100381610038161003816 lt 119863120573
119899 (20)
In the formula 119863119899is a random variable whose distribu-
tion depends on 119899119863120573119899is critical value for a significant level 120573
It is considered that the distribution to be used at a significantlevel 120573 cannot be resisted otherwise it should be rejected
The distribution formwas tested through the K-Smethodto determine the probability density function Fuzzy warningindicator was then determinedwith different significant levelIn dam safety evaluation significant level 120572 is the probabilityof the dam failure Supposing 119878
119898is the extreme of the
information entropy of the upstream overall deformation if119878 gt 119878
119898 the probability of the dam failure is 119875(119878 gt 119878
119898) =
120572 = intinfin
119878119898
119891(119909)119889119909 and the reliability index of dam failure is1 minus 120572 According to the dam importance different failureprobability is set and the multistage warning indicators wereidentified
6 Example Analysis
61 Project Profile One flat-slab deck dam built with rein-forced concrete is an important part of one river basin cascadeexploitation The elevation of this dam crest is 13770m andthe height of biggest part is about 43m the crest runs 2250min length and is made of 27 flat-slab buttresses with a span of75m The space between the left side of 2 buttress and theright side of 9 buttress is the joint part the overflow buttressis located from the 9 buttress to the 20 buttress the rest isthe water-retaining buttressTheworkshop buttress is locatedfrom the 5 buttress to the 8 buttress In this dam the levelof deadwater is 1220m the normal highwater level is 1310m(in practice it is 1290m) the design flood level is 1367m andthe check flood level is 1375m To monitor the displacementof this dam a direct plumb line and an inverted plumb linewere arranged in four buttresses 4 9 21 and 24There isa crushed zone under the dam foundation where occurrenceis N20∘sim25∘W SWang70∘sim80∘ the maximum width is about3m and the narrowest place is about 1mThere is an elevationclip joint mud at level 91m
The deformation field characterized by the observationpoint at 21 should be typical and is a key point because it isin the riverbedThus the observation point G21 at the heightof 134m along the direct plumb line at 21 was analyzed aswell as point 27 at the height of 118m along this line andpoint 28 at the height of 107mAll these valuesmeasuredweretransformed into absolute displacement The arrangement ofeach point is shown in Figure 8 The daily monitoring dataseries is from January 1 2003 to December 31 2013
In this paper two-stage warning indicators were setaccording to the practical running of this project and danger120572 = 5 is the primary warning that is mainly used to discri-minate and handle early dangerous case and the reliabilityindex of dam failure is 95 whereas 120572 = 1 is the secondarywarning that is mainly used to determine grave danger andprevent urgent danger and the reliability index of dam failureis 99
8 Mathematical Problems in Engineering
G21 G9 G4G24
25
30
2631
28
27
3233
Figure 8 The arrangement of observation point
Ups
tream
wat
er le
vel
623
198
4
121
419
89
66
1995
112
620
00
519
200
6
119
201
1
11
1979
Date
120124128132
Figure 9 The process line of upstream water level
11
2007
11
2008
11
2006
123
120
08
11
2005
123
120
09
123
120
10
123
120
11
123
020
12
123
020
13
Data
05
101520253035
Tem
pera
ture
Figure 10 The process line of temperature
62 Calculating the Contribution of Deformation at theObservation Point to the Overall Deformation Figures 9-10 show the upstream stage hydrograph and temperaturestage hydrograph respectively The water stage remainedunchanged whereas the temperature changed in the annualcycle Figure 11 shows the relationship between informationentropy of overall deformation and temperature in 2007 Anegative correlation exists between the overall deformationof 21 buttress and temperature that is an increase in tem-perature corresponds to the decrease in upstream or down-stream displacement and a decrease in temperature meansan increase in the upstream or downstream displacementFigure 12 shows the correlation between the displacementat observation point G21 and the temperature The overalldeformation law is roughly identical with that at observationpoint G21
Figure 12 reveals that the temperature obviously influ-enced overall deformation that is the temperature canchange the contribution of single observation point to theoverall deformation Temperature change was divided intothe stage of temperature rise and stage of temperature drop
2011
11
2011
21
2011
31
2011
41
2011
51
2011
61
2011
71
2011
81
2011
91
2011
10
1
2011
11
1
2011
12
1
TemperatureDeformed entropy
0070
140210280
Tem
pera
ture
minus080minus076minus072minus068minus064minus060
Def
orm
ed en
tropy
Figure 11 The relationship between information entropy of overalldeformation and temperature in 2011
2011
11
2011
13
1
2011
32
2011
41
2011
51
2011
53
1
2011
63
0
2011
73
0
2011
82
9
2011
92
8
2011
10
28
2011
11
27
2011
12
27
Horizontal displacementTemperature
05
101520253035
Tem
pera
ture
minus16minus12minus08minus04004
Hor
izon
tal
disp
lace
men
t
Figure 12 The relationship between horizontal displacement andtemperature of observation point G21 in 2011
The weight of deformation at observation point in two stageswas calculated The results are shown in the Table 1
63 Results of Information Entropy of Overall DeformationRange of cloud fall of the displacement at G21 27 and28 is shown in Figure 13 The boundary values of mostdangerous information entropy from 2003 to 2013 are shownin Tables 2ndash5The absolute value means the degree of dangerdownstream is set as negative value In the significance level120572 = 005 7 kinds of common distribution (lognormal distri-bution normal distribution uniform distribution triangulardistribution exponential distribution 120574 distribution and 120573distribution) were used to carry on hypothesis test for 1198780
119898119895
1198781
119898119895 1198770119898119895 and 1198771
119898119895 with K-S method The maximum 119863
119899
of each sequencewas obtained and comparedwith the criticalvalue 119863005
119899of the significant level 005 to determine the type
Mathematical Problems in Engineering 9
Table 1 The weight table of observation point
Observation point Altitude The period of temperature rise The period of temperature dropG21 134m 0357 039227 118m 0331 031628 107m 0312 0292
Table 2 The lower limit of the most dangerous information entropy of downstream overall deformation
Year 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013Information entropy 04784 04837 04767 04784 04969 04887 04969 05102 05082 04799 04739
Table 3 The upper limit of the most dangerous information entropy of downstream overall deformation
Year 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013Information entropy 04791 04845 04770 04786 04971 04899 04974 05186 05161 04817 04769
Table 4 The lower limit of the most dangerous information entropy of upstream overall deformation
Year 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013Information entropy minus05433 minus05491 minus04773 minus05083 minus05067 minus04784 minus05080 minus04839 minus04836 minus04825 minus05020
Table 5 The lower limit of the most dangerous information entropy of upstream overall deformation
Year 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013Information entropy minus05339 minus05440 minus04770 minus05079 minus05055 minus04770 minus05040 minus04822 minus04806 minus04754 minus04766
Cloud droplets
minus15
minus14
minus13
minus12
minus11
minus1
minus09
minus08
minus07
minus06
minus05
minus04
minus03
minus02
minus01
010
02
03
04
05
06
07
08
09 1
00102030405060708091
Cer
tain
ty
G21 27 28
Figure 13 Range of cloud fall of the displacement at G21 27 and 28
of the best distribution K-S test results are shown in Table 6Multistage fuzzy warning values are presented in Table 7
K-S test shows that 1198780119898119895 1198781119898119895 1198770119898119895 and 1198771
119898119895 satisfy
normal distributionThe probability density function of the sequence is
119891 (119909) =1
radic21205871205902exp(minus
(119909 minus 120583)2
120590) (21)
Parameter values of (21) are presented in Table 7In downstream deformation if 120572 = 5 the primary
warning indicator is (04763 04775) if 120572 = 1 thesecondary warning indicator is (04757 04763) 120575lowast is used as
the boundary value when two indicators overlap In the caseof upstream deformation if 120572 = 5 the primary warningindicator is (minus04779 minus04768) if 120572 = 1 the secondarywarning indicator is (minus04768 minus04763) (Table 8)
64 The Structure Calculation of Monitoring Index of DamHorizontal Displacement According to the actual situationthree-dimensional finite element model of the dam is estab-lished According to the structure and basic geologicalconditions of 21 dam section the scope of finite elementcalculation model can be got as follows taking 15 times ashigh dam in the upstream direction taking 15 times as highdam in the upstream direction and taking 1 time as highdam below the dam foundation The model is constitutedof 11445 nodes and 8571 units The unit type is 6 sides 8nodes isoparametric element The deformation observationdata analysis shows the dam under the condition of lowtemperature and highwater level there is larger displacementin the downstream when in high temperature and in lowwater level there is larger displacement in the upstream(Table 9 Figure 14) In view of the actual situation of thedam observation and data analysis results the load conditionselection is as follows (Table 10 Figures 15 and 16)
Working condition 1 normal water level 1290m andmaximum temperature dropWorking condition 2 dead water level 1220m andmaximum temperature rise
10 Mathematical Problems in Engineering
Table 6 K-S test results
Probability distribution 1198780
1198981198951198781
1198981198951198770
1198981198951198771
119898119895
Lognormal distribution 026 011 023 028Normal distribution 011 008 026 022Uniform distribution 074 155 088 068Triangular distribution 053 054 059 061Exponential distribution 041 042 055 035120574 distribution 037 036 031 033120573 distribution 068 073 086 063119863005
119899034 034 029 029
The most reasonable probability distribution Normal distribution Normal distribution Normal distribution Normal distribution
Table 7 Parameter values of the probability density function
Data series Parameter values120583 120590
2
1198780
119898119895 0488355 00129
1198781
119898119895 0490627 00151
1198770
119898119895 minus050210 00249
1198771
119898119895 minus049674 00245
Figure 14 Finite element model of the dam
The primary warning indicators of concrete dam wereobtained by calculation methods for structures If the infor-mation entropy of the overall deformation reached 04770the dammoveddownstreamand in the state of primarywarn-ing If the information entropy of the overall deformationreached minus04773 the dam moved upstream and in the stateof primary warning (Table 11) They all fell into intervalscalculated by fuzzy methodologyThe analysis shows that themethod brought up in this paper is reasonable and scientificAlso the analysis shows the physical meaning of the fuzzywarning indexUnder the action of the unfavorable load com-bination and the influence of the complex random factorsthe maximum entropy andminimum information entropy ofthe overall deformation lie in this interval Considering theinfluences of random factorsmultistage fuzzywarning valueswere receptive and safe
In this paper the overall deformation of 21 buttresswas analyzed through the theoretical method proposedInfluences of random factors on the warning value wereconsidered and multistage fuzzy warning values were deter-mined If the information entropy of overall deformationis in (04763 04775) the dam moved downstream and in
minus2399e minus 003minus1296e minus 003minus1940e minus 0049084e minus 0042011e minus 0033113e minus 0034216e minus 0035318e minus 0036421e minus 0037523e minus 0038625e minus 003
XY
Z
Figure 15The results of finite element calculation ofworking condi-tion 1
XY
Z minus6007e minus 004minus5014e minus 004minus4021e minus 004minus3028e minus 004minus2035e minus 004minus1042e minus 004minus4872e minus 0069443e minus 0051937e minus 0042930e minus 0043923e minus 004
Figure 16 The results of finite element calculation of workingcondition 2
the state of primary warning If the information entropy ofoverall deformation is in (04757 04763) the dam moveddownstream and in the state of secondary warning If theinformation entropy of overall deformation is in (minus04779minus04768) the dam moved upstream and in the state ofprimary warning If the information entropy of the over-all deformation is in (minus04768 minus04763) the dam movedupstream and in the state of secondary warning
7 Conclusion
This paper presented multistage warning indicators of con-crete dam space and considered the influences of complex
Mathematical Problems in Engineering 11
Table 8 Multistage fuzzy warning values of concrete dam
The direction of deformation Confidence levelThe primary warning indicator The secondary warning indicator
Downstream (04763 04775) (04757 04763)Upstream (minus04779 minus04768) (minus04768 minus04763)
Table 9 Material parameter of the dam
Structure Density (kgm) Poissonrsquos ratio Elastic modulus (GPa)Concrete face slab 2400 0167 24Buttress 2400 0167 24Partition wall 2400 0167 24Reinforced concrete block 2400 0160 22Foundation rock mass 2700 0175 12Diorite-dyke 2000 03 115Crushed zone 2000 03 029Horizontal joints 2000 03 07
Table 10 The results of finite element calculation of displacementof observation point (mm)
Observation point Working condition 1 Working condition 2G21 0785 minus059927 0654 minus038628 0356 0114
Table 11 The primary warning indicators of concrete dam
The direction of deformation The primary warning indicatorDownstream 04770Upstream minus04773
random factors The results of the specific studies are asfollows
(1) Influences of fuzziness and randomness of randomfactors on the long-term service of dam were dis-cussed a fuzziness indicator that measures the influ-ence of random factors on monitoring value wasconstructed through cloud model
(2) Equivalent model of overall deformation was pro-posed In the model the overall deformation ofdam was regarded as a deformation system whereeach observation point had different contributions(weights) and affected one another Based on entropytheory a space information entropy that can measurethe overall deformation condition was established
(3) Multistage warning indicators against overall defor-mation of concrete dam under the influences of fuzzi-ness and randomness were determined and nondeter-ministic optimal control of the indicator was achievedto improve the competence of warning against thedeformation of concrete dam
Competing Interests
No conflict of interests exits in the submission of this paper
Acknowledgments
This paper was financially supported by National Natu-ral Science Foundation of China (Grants nos 4132300151139001 51579086 51379068 51279052 51579083 and51209077) Jiangsu Natural Science Foundation (Grants nosBK20140039 and BK2012036) Research Fund for the Doc-toral Program of Higher Education of China (Grant no20130094110010) the Ministry of Water Resources PublicWelfare Industry Research Special Fund Project (Grantsnos 201201038 and 201301061) Jiangsu Province ldquo333 High-Level Personnel Training Projectrdquo (Grants nos BRA2011179and BRA2011145) Project Funded by the Priority AcademicProgram Development of Jiangsu Higher Education Institu-tions (Grant no YS11001) Jiangsu Province ldquo333 High-LevelPersonnel Training Projectrdquo (Grant no 2017-B08037) JiangsuProvince ldquoSix Talent Peaksrdquo Project (Grant no JY-008) andFundamental Research Funds for the Central Universities(Grants nos 2016B04114 and 2015B25414)
References
[1] F BenedettoGGiunta andLMastroeni ldquoAmaximumentropymethod to assess the predictability of financial and commoditypricesrdquo Digital Signal Processing vol 46 pp 19ndash31 2015
[2] S Muraro A Madaschi and A Gajo ldquoOn the reliability of 3Dnumerical analyses on passive piles used for slope stabilisationin frictional soilsrdquo Geotechnique vol 64 no 6 pp 486ndash4922014
[3] W J Yoon and K S Park ldquoA study on the market instabilityindex and risk warning levels in early warning system foreconomic crisisrdquo Digital Signal Processing vol 29 pp 35ndash442014
[4] D Tonini ldquoObserved behavior of several leakier arch damsrdquoJournal of the Power Division vol 82 no 12 pp 135ndash139 1956
12 Mathematical Problems in Engineering
[5] R Salgado and D Kim ldquoReliability analysis of load andresistance factor design of slopesrdquo Journal of Geotechnical andGeoenvironmental Engineering vol 140 no 1 pp 57ndash73 2014
[6] Y C Gu and S J Wang ldquoStudy on the early-warning thresholdof structural instability unexpected accidents of reservoir damrdquoJournal of Hydraulic Engineering vol 40 no 12 pp 1467ndash14722009
[7] H Z Su F Wang and H P Liu ldquoEarly-warning index for damservice behavior based on POT modelrdquo Journal of HydraulicEngineering vol 43 no 8 pp 974ndash986 2012
[8] L Chen Mechanics performance with parameters changing inspace amp monitoring model of roller compacted concrete dam[PhD thesis] Hohai University 2006
[9] A B Li L F Zhang Q L Wang B Li Z Li and Y WangldquoInformation theory in nonlinear error growth dynamics andits application to predictability taking the Lorenz system as anexamplerdquo Science China Earth Sciences vol 56 no 8 pp 1413ndash1421 2013
[10] O Victor ldquoThe theory of cloudsrdquo Library Journal vol 132 no13 p 62 2007
[11] M Yang H Z Su and X Q Yan ldquoComputation and analysis ofhigh rocky slope safety in a water conservancy projectrdquoDiscreteDynamics in Nature and Society vol 2015 Article ID 197579 11pages 2015
[12] C Ricotta and G C Avena ldquoEvaluating the degree of fuzzi-ness of thematic maps with a generalized entropy functiona methodological outlookrdquo International Journal of RemoteSensing vol 23 no 20 pp 4519ndash4523 2002
[13] V I Khvorostyanov and J A Curry ldquoParameterization ofhomogeneous ice nucleation for cloud and climate modelsbased on classical nucleation theoryrdquo Atmospheric Chemistryand Physics vol 12 no 19 pp 9275ndash9302 2012
[14] H Z Su Z P Wen and Z R Wu ldquoStudy on an intelligentinference engine in early-warning system of damhealthrdquoWaterResources Management vol 25 no 6 pp 1545ndash1563 2011
[15] M Kaminski ldquoProbabilistic entropy in homogenization ofthe periodic fiber-reinforced composites with random elasticparametersrdquo International Journal for Numerical Methods inEngineering vol 90 no 8 pp 939ndash954 2012
[16] E Czogała and J Łeski ldquoApplication of entropy and energymeasures of fuzziness to processing of ECG signalrdquo Fuzzy Setsand Systems vol 97 no 1 pp 9ndash18 1998
[17] K Wu and J L Jin ldquoProjection pursuit model for evaluation ofregion water resource security based on changeable weight andinformationrdquo Resources and Environment in the Yangtze Basinvol 20 no 9 pp 1085ndash1091 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
Choice of sample
Input sample eigenvalues
Calling cloud model program
Generating range of cloud fall
Calculating the fuzzinessof all measured value
Constructing deformed entropyof one measured value
Calculating weight of singleobservation point
Constructing fuzzy deformedentropy of overall deformation
Constructing fuzzy deformedentropy of one measured value
Ex EnHe
Figure 6 Computational process of fuzzy information entropy of overall deformation
= minus
119899
sum
119894=1
120596119894ln120596119894
2
sum
119896=1
120583119896
119894119895minus
119899
sum
119894=1
120596119894
2
sum
119896=1
120583119896
119894119895ln 120583119896119894119895
= minus
119899
sum
119894=1
120596119894ln120596119894+
119899
sum
119894=1
120596119894119878119894119895
(15)
When the dam move deforms upstream the expressionof the information entropy of overall deformation is
119878119895=
119898
sum
119894=1
2
sum
119896=1
120596119894120583119896
119894119895ln (120596119894120583119896
119894119895) =
119899
sum
119894=1
120596119894ln120596119894+
119899
sum
119894=1
120596119894119878119894119895 (16)
Therefore the expression of information entropy of over-all deformation is defined as follows
119878119895=
minus
119899
sum
119894=1
120596119894ln120596119894+
119899
sum
119894=1
120596119894119878119894119895 119878119894119895ge 0
119899
sum
119894=1
120596119894ln120596119894+
119899
sum
119894=1
120596119894119878119894119895 119878
119894119895lt 0
(17)
The absolute value of the information entropy of the over-all deformationmeasures the danger level of the deformationthat is a smaller absolute value means a higher danger levelpositive and negative values stand for the direction of thedeformation a positive value means downstream deforma-tion whereas a negative value means upstream deformation
Considering the influences of random factors the fuzzyinformation entropy of 119909
119894119895is [1198780119894119895 1198781
119894119895] the fuzzy information
entropy of overall deformation can be illustrated through(17) Computational process of fuzzy information entropy ofoverall deformation is shown in Figure 6
5 Proposed Multistage FuzzyInformation Entropy of OverallDeformation Warning Indicators
Horizontal displacement of dam crest changes in an annualcycle ldquoupstream and downstream switchrdquo Therefore thisdisplacement should be in a certain scale and be controlledunder somemonitoring indicators for the safe damoperation
In the case of downstream displacement the primaryfuzzy warning indicator 1205751015840
1is defined as 1205751015840
1= (1205750
1 1205751
1) 12057501is
the lower limit of this indicator and 12057511is the upper limit
the secondary indicator 12057510158402is 12057510158402= (1205750
2 1205751
2) where 1205750
2is
the lower limit of this indicator and 12057512is the upper limit
When 12057511gt 1205750
2 a cross phenomenon appears in the primary
indicator and secondary indicator when both of them shouldbe categorized according to the membership of displacementmeasured 120575lowast was introduced because the membership ofdisplacement at this point is the same Figure 7 showsdiagram of multistage fuzzy information entropy warningindicators
The primary fuzzy warning indicator is 12057510158401= (1205750
1 120575lowast
) andthe secondary is 1205751015840
2= (120575lowast
1205751
2)
If the deformation value is in (12057501 1205751
1) or (1205750
1 120575lowast
) the damis in the state of primary warning if the value is in (1205750
2 1205751
2) or
(1205750
2 120575lowast
) the dam is in the state of secondary warningThe time sequence of deformation at each observation
point was analyzed by using the above theoretical methodThe lower and upper limits of the fuzzy information entropyof overall deformation affected by Δ
119894119895will be 1198780
119895 and 1198781
119895
Considering the damrsquos long-term service when the dammoves downstream the lower limit 1198780
119898119895 and upper limit
Mathematical Problems in Engineering 7
1Degree of membership
Displacement
1Degree of membership
Displacement
12057512120575021205751112057501
1205751212057502 1205751112057501 120575lowast
Figure 7Diagramofmultistage fuzzy information entropywarningindicators
1198781
119898119895 were selected and when the dam moves upstream
the lower limit 1198770119898119895 and upper limit 1198771
119898119895 were selected
1198780
119898119895 1198781119898119895 1198770119898119895 and 1198771
119898119895 are random variables and four
subsample spaces with the sample size of 119873 can be obtainedby the following
1198780
= 1198780
1198981 1198780
1198982 119878
0
119898119898
1198781
= 1198781
1198981 1198781
1198982 119878
1
119898119898
1198770
= 1198770
1198981 1198770
1198982 119877
0
119898119898
1198771
= 1198771
1198981 1198771
1198982 119877
1
119898119898
(18)
Shapiro-Wilk test andKolmogorov-Smirnov test can bothtest whether the samples obey normal distribution or not Butthe Kolmogorov-Smirnov test is applicable to fewer samplesIt can not only test if the samples are subject to normaldistribution but also test if samples are subject to otherdistributions The basic idea of the K-S test is to comparethe cumulative frequency of the observed value (119865
119899(119909)) with
the assumed theoretical probability distribution (119865119909(119909)) to
construct statisticsAccording to the method of empirical distribution func-
tion segmented cumulative frequency is obtained by usingthe following formula
119865119899(119909) =
0 119909 lt 119909119894
119894
119899 119909119894le 119909 lt 119909
119894+1
1 119909 ge 119909
(19)
In the formula 1199091 1199092 119909
119899is sample data after arrange-
ment The sample size is 119899
In the full range of random variable 119883 the maximumdifference between 119865
119899(119909) and 119865
119909(119909) is
119863119899= max 1003816100381610038161003816119865119909 (119909) minus 119865119899 (119909)
1003816100381610038161003816 lt 119863120573
119899 (20)
In the formula 119863119899is a random variable whose distribu-
tion depends on 119899119863120573119899is critical value for a significant level 120573
It is considered that the distribution to be used at a significantlevel 120573 cannot be resisted otherwise it should be rejected
The distribution formwas tested through the K-Smethodto determine the probability density function Fuzzy warningindicator was then determinedwith different significant levelIn dam safety evaluation significant level 120572 is the probabilityof the dam failure Supposing 119878
119898is the extreme of the
information entropy of the upstream overall deformation if119878 gt 119878
119898 the probability of the dam failure is 119875(119878 gt 119878
119898) =
120572 = intinfin
119878119898
119891(119909)119889119909 and the reliability index of dam failure is1 minus 120572 According to the dam importance different failureprobability is set and the multistage warning indicators wereidentified
6 Example Analysis
61 Project Profile One flat-slab deck dam built with rein-forced concrete is an important part of one river basin cascadeexploitation The elevation of this dam crest is 13770m andthe height of biggest part is about 43m the crest runs 2250min length and is made of 27 flat-slab buttresses with a span of75m The space between the left side of 2 buttress and theright side of 9 buttress is the joint part the overflow buttressis located from the 9 buttress to the 20 buttress the rest isthe water-retaining buttressTheworkshop buttress is locatedfrom the 5 buttress to the 8 buttress In this dam the levelof deadwater is 1220m the normal highwater level is 1310m(in practice it is 1290m) the design flood level is 1367m andthe check flood level is 1375m To monitor the displacementof this dam a direct plumb line and an inverted plumb linewere arranged in four buttresses 4 9 21 and 24There isa crushed zone under the dam foundation where occurrenceis N20∘sim25∘W SWang70∘sim80∘ the maximum width is about3m and the narrowest place is about 1mThere is an elevationclip joint mud at level 91m
The deformation field characterized by the observationpoint at 21 should be typical and is a key point because it isin the riverbedThus the observation point G21 at the heightof 134m along the direct plumb line at 21 was analyzed aswell as point 27 at the height of 118m along this line andpoint 28 at the height of 107mAll these valuesmeasuredweretransformed into absolute displacement The arrangement ofeach point is shown in Figure 8 The daily monitoring dataseries is from January 1 2003 to December 31 2013
In this paper two-stage warning indicators were setaccording to the practical running of this project and danger120572 = 5 is the primary warning that is mainly used to discri-minate and handle early dangerous case and the reliabilityindex of dam failure is 95 whereas 120572 = 1 is the secondarywarning that is mainly used to determine grave danger andprevent urgent danger and the reliability index of dam failureis 99
8 Mathematical Problems in Engineering
G21 G9 G4G24
25
30
2631
28
27
3233
Figure 8 The arrangement of observation point
Ups
tream
wat
er le
vel
623
198
4
121
419
89
66
1995
112
620
00
519
200
6
119
201
1
11
1979
Date
120124128132
Figure 9 The process line of upstream water level
11
2007
11
2008
11
2006
123
120
08
11
2005
123
120
09
123
120
10
123
120
11
123
020
12
123
020
13
Data
05
101520253035
Tem
pera
ture
Figure 10 The process line of temperature
62 Calculating the Contribution of Deformation at theObservation Point to the Overall Deformation Figures 9-10 show the upstream stage hydrograph and temperaturestage hydrograph respectively The water stage remainedunchanged whereas the temperature changed in the annualcycle Figure 11 shows the relationship between informationentropy of overall deformation and temperature in 2007 Anegative correlation exists between the overall deformationof 21 buttress and temperature that is an increase in tem-perature corresponds to the decrease in upstream or down-stream displacement and a decrease in temperature meansan increase in the upstream or downstream displacementFigure 12 shows the correlation between the displacementat observation point G21 and the temperature The overalldeformation law is roughly identical with that at observationpoint G21
Figure 12 reveals that the temperature obviously influ-enced overall deformation that is the temperature canchange the contribution of single observation point to theoverall deformation Temperature change was divided intothe stage of temperature rise and stage of temperature drop
2011
11
2011
21
2011
31
2011
41
2011
51
2011
61
2011
71
2011
81
2011
91
2011
10
1
2011
11
1
2011
12
1
TemperatureDeformed entropy
0070
140210280
Tem
pera
ture
minus080minus076minus072minus068minus064minus060
Def
orm
ed en
tropy
Figure 11 The relationship between information entropy of overalldeformation and temperature in 2011
2011
11
2011
13
1
2011
32
2011
41
2011
51
2011
53
1
2011
63
0
2011
73
0
2011
82
9
2011
92
8
2011
10
28
2011
11
27
2011
12
27
Horizontal displacementTemperature
05
101520253035
Tem
pera
ture
minus16minus12minus08minus04004
Hor
izon
tal
disp
lace
men
t
Figure 12 The relationship between horizontal displacement andtemperature of observation point G21 in 2011
The weight of deformation at observation point in two stageswas calculated The results are shown in the Table 1
63 Results of Information Entropy of Overall DeformationRange of cloud fall of the displacement at G21 27 and28 is shown in Figure 13 The boundary values of mostdangerous information entropy from 2003 to 2013 are shownin Tables 2ndash5The absolute value means the degree of dangerdownstream is set as negative value In the significance level120572 = 005 7 kinds of common distribution (lognormal distri-bution normal distribution uniform distribution triangulardistribution exponential distribution 120574 distribution and 120573distribution) were used to carry on hypothesis test for 1198780
119898119895
1198781
119898119895 1198770119898119895 and 1198771
119898119895 with K-S method The maximum 119863
119899
of each sequencewas obtained and comparedwith the criticalvalue 119863005
119899of the significant level 005 to determine the type
Mathematical Problems in Engineering 9
Table 1 The weight table of observation point
Observation point Altitude The period of temperature rise The period of temperature dropG21 134m 0357 039227 118m 0331 031628 107m 0312 0292
Table 2 The lower limit of the most dangerous information entropy of downstream overall deformation
Year 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013Information entropy 04784 04837 04767 04784 04969 04887 04969 05102 05082 04799 04739
Table 3 The upper limit of the most dangerous information entropy of downstream overall deformation
Year 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013Information entropy 04791 04845 04770 04786 04971 04899 04974 05186 05161 04817 04769
Table 4 The lower limit of the most dangerous information entropy of upstream overall deformation
Year 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013Information entropy minus05433 minus05491 minus04773 minus05083 minus05067 minus04784 minus05080 minus04839 minus04836 minus04825 minus05020
Table 5 The lower limit of the most dangerous information entropy of upstream overall deformation
Year 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013Information entropy minus05339 minus05440 minus04770 minus05079 minus05055 minus04770 minus05040 minus04822 minus04806 minus04754 minus04766
Cloud droplets
minus15
minus14
minus13
minus12
minus11
minus1
minus09
minus08
minus07
minus06
minus05
minus04
minus03
minus02
minus01
010
02
03
04
05
06
07
08
09 1
00102030405060708091
Cer
tain
ty
G21 27 28
Figure 13 Range of cloud fall of the displacement at G21 27 and 28
of the best distribution K-S test results are shown in Table 6Multistage fuzzy warning values are presented in Table 7
K-S test shows that 1198780119898119895 1198781119898119895 1198770119898119895 and 1198771
119898119895 satisfy
normal distributionThe probability density function of the sequence is
119891 (119909) =1
radic21205871205902exp(minus
(119909 minus 120583)2
120590) (21)
Parameter values of (21) are presented in Table 7In downstream deformation if 120572 = 5 the primary
warning indicator is (04763 04775) if 120572 = 1 thesecondary warning indicator is (04757 04763) 120575lowast is used as
the boundary value when two indicators overlap In the caseof upstream deformation if 120572 = 5 the primary warningindicator is (minus04779 minus04768) if 120572 = 1 the secondarywarning indicator is (minus04768 minus04763) (Table 8)
64 The Structure Calculation of Monitoring Index of DamHorizontal Displacement According to the actual situationthree-dimensional finite element model of the dam is estab-lished According to the structure and basic geologicalconditions of 21 dam section the scope of finite elementcalculation model can be got as follows taking 15 times ashigh dam in the upstream direction taking 15 times as highdam in the upstream direction and taking 1 time as highdam below the dam foundation The model is constitutedof 11445 nodes and 8571 units The unit type is 6 sides 8nodes isoparametric element The deformation observationdata analysis shows the dam under the condition of lowtemperature and highwater level there is larger displacementin the downstream when in high temperature and in lowwater level there is larger displacement in the upstream(Table 9 Figure 14) In view of the actual situation of thedam observation and data analysis results the load conditionselection is as follows (Table 10 Figures 15 and 16)
Working condition 1 normal water level 1290m andmaximum temperature dropWorking condition 2 dead water level 1220m andmaximum temperature rise
10 Mathematical Problems in Engineering
Table 6 K-S test results
Probability distribution 1198780
1198981198951198781
1198981198951198770
1198981198951198771
119898119895
Lognormal distribution 026 011 023 028Normal distribution 011 008 026 022Uniform distribution 074 155 088 068Triangular distribution 053 054 059 061Exponential distribution 041 042 055 035120574 distribution 037 036 031 033120573 distribution 068 073 086 063119863005
119899034 034 029 029
The most reasonable probability distribution Normal distribution Normal distribution Normal distribution Normal distribution
Table 7 Parameter values of the probability density function
Data series Parameter values120583 120590
2
1198780
119898119895 0488355 00129
1198781
119898119895 0490627 00151
1198770
119898119895 minus050210 00249
1198771
119898119895 minus049674 00245
Figure 14 Finite element model of the dam
The primary warning indicators of concrete dam wereobtained by calculation methods for structures If the infor-mation entropy of the overall deformation reached 04770the dammoveddownstreamand in the state of primarywarn-ing If the information entropy of the overall deformationreached minus04773 the dam moved upstream and in the stateof primary warning (Table 11) They all fell into intervalscalculated by fuzzy methodologyThe analysis shows that themethod brought up in this paper is reasonable and scientificAlso the analysis shows the physical meaning of the fuzzywarning indexUnder the action of the unfavorable load com-bination and the influence of the complex random factorsthe maximum entropy andminimum information entropy ofthe overall deformation lie in this interval Considering theinfluences of random factorsmultistage fuzzywarning valueswere receptive and safe
In this paper the overall deformation of 21 buttresswas analyzed through the theoretical method proposedInfluences of random factors on the warning value wereconsidered and multistage fuzzy warning values were deter-mined If the information entropy of overall deformationis in (04763 04775) the dam moved downstream and in
minus2399e minus 003minus1296e minus 003minus1940e minus 0049084e minus 0042011e minus 0033113e minus 0034216e minus 0035318e minus 0036421e minus 0037523e minus 0038625e minus 003
XY
Z
Figure 15The results of finite element calculation ofworking condi-tion 1
XY
Z minus6007e minus 004minus5014e minus 004minus4021e minus 004minus3028e minus 004minus2035e minus 004minus1042e minus 004minus4872e minus 0069443e minus 0051937e minus 0042930e minus 0043923e minus 004
Figure 16 The results of finite element calculation of workingcondition 2
the state of primary warning If the information entropy ofoverall deformation is in (04757 04763) the dam moveddownstream and in the state of secondary warning If theinformation entropy of overall deformation is in (minus04779minus04768) the dam moved upstream and in the state ofprimary warning If the information entropy of the over-all deformation is in (minus04768 minus04763) the dam movedupstream and in the state of secondary warning
7 Conclusion
This paper presented multistage warning indicators of con-crete dam space and considered the influences of complex
Mathematical Problems in Engineering 11
Table 8 Multistage fuzzy warning values of concrete dam
The direction of deformation Confidence levelThe primary warning indicator The secondary warning indicator
Downstream (04763 04775) (04757 04763)Upstream (minus04779 minus04768) (minus04768 minus04763)
Table 9 Material parameter of the dam
Structure Density (kgm) Poissonrsquos ratio Elastic modulus (GPa)Concrete face slab 2400 0167 24Buttress 2400 0167 24Partition wall 2400 0167 24Reinforced concrete block 2400 0160 22Foundation rock mass 2700 0175 12Diorite-dyke 2000 03 115Crushed zone 2000 03 029Horizontal joints 2000 03 07
Table 10 The results of finite element calculation of displacementof observation point (mm)
Observation point Working condition 1 Working condition 2G21 0785 minus059927 0654 minus038628 0356 0114
Table 11 The primary warning indicators of concrete dam
The direction of deformation The primary warning indicatorDownstream 04770Upstream minus04773
random factors The results of the specific studies are asfollows
(1) Influences of fuzziness and randomness of randomfactors on the long-term service of dam were dis-cussed a fuzziness indicator that measures the influ-ence of random factors on monitoring value wasconstructed through cloud model
(2) Equivalent model of overall deformation was pro-posed In the model the overall deformation ofdam was regarded as a deformation system whereeach observation point had different contributions(weights) and affected one another Based on entropytheory a space information entropy that can measurethe overall deformation condition was established
(3) Multistage warning indicators against overall defor-mation of concrete dam under the influences of fuzzi-ness and randomness were determined and nondeter-ministic optimal control of the indicator was achievedto improve the competence of warning against thedeformation of concrete dam
Competing Interests
No conflict of interests exits in the submission of this paper
Acknowledgments
This paper was financially supported by National Natu-ral Science Foundation of China (Grants nos 4132300151139001 51579086 51379068 51279052 51579083 and51209077) Jiangsu Natural Science Foundation (Grants nosBK20140039 and BK2012036) Research Fund for the Doc-toral Program of Higher Education of China (Grant no20130094110010) the Ministry of Water Resources PublicWelfare Industry Research Special Fund Project (Grantsnos 201201038 and 201301061) Jiangsu Province ldquo333 High-Level Personnel Training Projectrdquo (Grants nos BRA2011179and BRA2011145) Project Funded by the Priority AcademicProgram Development of Jiangsu Higher Education Institu-tions (Grant no YS11001) Jiangsu Province ldquo333 High-LevelPersonnel Training Projectrdquo (Grant no 2017-B08037) JiangsuProvince ldquoSix Talent Peaksrdquo Project (Grant no JY-008) andFundamental Research Funds for the Central Universities(Grants nos 2016B04114 and 2015B25414)
References
[1] F BenedettoGGiunta andLMastroeni ldquoAmaximumentropymethod to assess the predictability of financial and commoditypricesrdquo Digital Signal Processing vol 46 pp 19ndash31 2015
[2] S Muraro A Madaschi and A Gajo ldquoOn the reliability of 3Dnumerical analyses on passive piles used for slope stabilisationin frictional soilsrdquo Geotechnique vol 64 no 6 pp 486ndash4922014
[3] W J Yoon and K S Park ldquoA study on the market instabilityindex and risk warning levels in early warning system foreconomic crisisrdquo Digital Signal Processing vol 29 pp 35ndash442014
[4] D Tonini ldquoObserved behavior of several leakier arch damsrdquoJournal of the Power Division vol 82 no 12 pp 135ndash139 1956
12 Mathematical Problems in Engineering
[5] R Salgado and D Kim ldquoReliability analysis of load andresistance factor design of slopesrdquo Journal of Geotechnical andGeoenvironmental Engineering vol 140 no 1 pp 57ndash73 2014
[6] Y C Gu and S J Wang ldquoStudy on the early-warning thresholdof structural instability unexpected accidents of reservoir damrdquoJournal of Hydraulic Engineering vol 40 no 12 pp 1467ndash14722009
[7] H Z Su F Wang and H P Liu ldquoEarly-warning index for damservice behavior based on POT modelrdquo Journal of HydraulicEngineering vol 43 no 8 pp 974ndash986 2012
[8] L Chen Mechanics performance with parameters changing inspace amp monitoring model of roller compacted concrete dam[PhD thesis] Hohai University 2006
[9] A B Li L F Zhang Q L Wang B Li Z Li and Y WangldquoInformation theory in nonlinear error growth dynamics andits application to predictability taking the Lorenz system as anexamplerdquo Science China Earth Sciences vol 56 no 8 pp 1413ndash1421 2013
[10] O Victor ldquoThe theory of cloudsrdquo Library Journal vol 132 no13 p 62 2007
[11] M Yang H Z Su and X Q Yan ldquoComputation and analysis ofhigh rocky slope safety in a water conservancy projectrdquoDiscreteDynamics in Nature and Society vol 2015 Article ID 197579 11pages 2015
[12] C Ricotta and G C Avena ldquoEvaluating the degree of fuzzi-ness of thematic maps with a generalized entropy functiona methodological outlookrdquo International Journal of RemoteSensing vol 23 no 20 pp 4519ndash4523 2002
[13] V I Khvorostyanov and J A Curry ldquoParameterization ofhomogeneous ice nucleation for cloud and climate modelsbased on classical nucleation theoryrdquo Atmospheric Chemistryand Physics vol 12 no 19 pp 9275ndash9302 2012
[14] H Z Su Z P Wen and Z R Wu ldquoStudy on an intelligentinference engine in early-warning system of damhealthrdquoWaterResources Management vol 25 no 6 pp 1545ndash1563 2011
[15] M Kaminski ldquoProbabilistic entropy in homogenization ofthe periodic fiber-reinforced composites with random elasticparametersrdquo International Journal for Numerical Methods inEngineering vol 90 no 8 pp 939ndash954 2012
[16] E Czogała and J Łeski ldquoApplication of entropy and energymeasures of fuzziness to processing of ECG signalrdquo Fuzzy Setsand Systems vol 97 no 1 pp 9ndash18 1998
[17] K Wu and J L Jin ldquoProjection pursuit model for evaluation ofregion water resource security based on changeable weight andinformationrdquo Resources and Environment in the Yangtze Basinvol 20 no 9 pp 1085ndash1091 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 7
1Degree of membership
Displacement
1Degree of membership
Displacement
12057512120575021205751112057501
1205751212057502 1205751112057501 120575lowast
Figure 7Diagramofmultistage fuzzy information entropywarningindicators
1198781
119898119895 were selected and when the dam moves upstream
the lower limit 1198770119898119895 and upper limit 1198771
119898119895 were selected
1198780
119898119895 1198781119898119895 1198770119898119895 and 1198771
119898119895 are random variables and four
subsample spaces with the sample size of 119873 can be obtainedby the following
1198780
= 1198780
1198981 1198780
1198982 119878
0
119898119898
1198781
= 1198781
1198981 1198781
1198982 119878
1
119898119898
1198770
= 1198770
1198981 1198770
1198982 119877
0
119898119898
1198771
= 1198771
1198981 1198771
1198982 119877
1
119898119898
(18)
Shapiro-Wilk test andKolmogorov-Smirnov test can bothtest whether the samples obey normal distribution or not Butthe Kolmogorov-Smirnov test is applicable to fewer samplesIt can not only test if the samples are subject to normaldistribution but also test if samples are subject to otherdistributions The basic idea of the K-S test is to comparethe cumulative frequency of the observed value (119865
119899(119909)) with
the assumed theoretical probability distribution (119865119909(119909)) to
construct statisticsAccording to the method of empirical distribution func-
tion segmented cumulative frequency is obtained by usingthe following formula
119865119899(119909) =
0 119909 lt 119909119894
119894
119899 119909119894le 119909 lt 119909
119894+1
1 119909 ge 119909
(19)
In the formula 1199091 1199092 119909
119899is sample data after arrange-
ment The sample size is 119899
In the full range of random variable 119883 the maximumdifference between 119865
119899(119909) and 119865
119909(119909) is
119863119899= max 1003816100381610038161003816119865119909 (119909) minus 119865119899 (119909)
1003816100381610038161003816 lt 119863120573
119899 (20)
In the formula 119863119899is a random variable whose distribu-
tion depends on 119899119863120573119899is critical value for a significant level 120573
It is considered that the distribution to be used at a significantlevel 120573 cannot be resisted otherwise it should be rejected
The distribution formwas tested through the K-Smethodto determine the probability density function Fuzzy warningindicator was then determinedwith different significant levelIn dam safety evaluation significant level 120572 is the probabilityof the dam failure Supposing 119878
119898is the extreme of the
information entropy of the upstream overall deformation if119878 gt 119878
119898 the probability of the dam failure is 119875(119878 gt 119878
119898) =
120572 = intinfin
119878119898
119891(119909)119889119909 and the reliability index of dam failure is1 minus 120572 According to the dam importance different failureprobability is set and the multistage warning indicators wereidentified
6 Example Analysis
61 Project Profile One flat-slab deck dam built with rein-forced concrete is an important part of one river basin cascadeexploitation The elevation of this dam crest is 13770m andthe height of biggest part is about 43m the crest runs 2250min length and is made of 27 flat-slab buttresses with a span of75m The space between the left side of 2 buttress and theright side of 9 buttress is the joint part the overflow buttressis located from the 9 buttress to the 20 buttress the rest isthe water-retaining buttressTheworkshop buttress is locatedfrom the 5 buttress to the 8 buttress In this dam the levelof deadwater is 1220m the normal highwater level is 1310m(in practice it is 1290m) the design flood level is 1367m andthe check flood level is 1375m To monitor the displacementof this dam a direct plumb line and an inverted plumb linewere arranged in four buttresses 4 9 21 and 24There isa crushed zone under the dam foundation where occurrenceis N20∘sim25∘W SWang70∘sim80∘ the maximum width is about3m and the narrowest place is about 1mThere is an elevationclip joint mud at level 91m
The deformation field characterized by the observationpoint at 21 should be typical and is a key point because it isin the riverbedThus the observation point G21 at the heightof 134m along the direct plumb line at 21 was analyzed aswell as point 27 at the height of 118m along this line andpoint 28 at the height of 107mAll these valuesmeasuredweretransformed into absolute displacement The arrangement ofeach point is shown in Figure 8 The daily monitoring dataseries is from January 1 2003 to December 31 2013
In this paper two-stage warning indicators were setaccording to the practical running of this project and danger120572 = 5 is the primary warning that is mainly used to discri-minate and handle early dangerous case and the reliabilityindex of dam failure is 95 whereas 120572 = 1 is the secondarywarning that is mainly used to determine grave danger andprevent urgent danger and the reliability index of dam failureis 99
8 Mathematical Problems in Engineering
G21 G9 G4G24
25
30
2631
28
27
3233
Figure 8 The arrangement of observation point
Ups
tream
wat
er le
vel
623
198
4
121
419
89
66
1995
112
620
00
519
200
6
119
201
1
11
1979
Date
120124128132
Figure 9 The process line of upstream water level
11
2007
11
2008
11
2006
123
120
08
11
2005
123
120
09
123
120
10
123
120
11
123
020
12
123
020
13
Data
05
101520253035
Tem
pera
ture
Figure 10 The process line of temperature
62 Calculating the Contribution of Deformation at theObservation Point to the Overall Deformation Figures 9-10 show the upstream stage hydrograph and temperaturestage hydrograph respectively The water stage remainedunchanged whereas the temperature changed in the annualcycle Figure 11 shows the relationship between informationentropy of overall deformation and temperature in 2007 Anegative correlation exists between the overall deformationof 21 buttress and temperature that is an increase in tem-perature corresponds to the decrease in upstream or down-stream displacement and a decrease in temperature meansan increase in the upstream or downstream displacementFigure 12 shows the correlation between the displacementat observation point G21 and the temperature The overalldeformation law is roughly identical with that at observationpoint G21
Figure 12 reveals that the temperature obviously influ-enced overall deformation that is the temperature canchange the contribution of single observation point to theoverall deformation Temperature change was divided intothe stage of temperature rise and stage of temperature drop
2011
11
2011
21
2011
31
2011
41
2011
51
2011
61
2011
71
2011
81
2011
91
2011
10
1
2011
11
1
2011
12
1
TemperatureDeformed entropy
0070
140210280
Tem
pera
ture
minus080minus076minus072minus068minus064minus060
Def
orm
ed en
tropy
Figure 11 The relationship between information entropy of overalldeformation and temperature in 2011
2011
11
2011
13
1
2011
32
2011
41
2011
51
2011
53
1
2011
63
0
2011
73
0
2011
82
9
2011
92
8
2011
10
28
2011
11
27
2011
12
27
Horizontal displacementTemperature
05
101520253035
Tem
pera
ture
minus16minus12minus08minus04004
Hor
izon
tal
disp
lace
men
t
Figure 12 The relationship between horizontal displacement andtemperature of observation point G21 in 2011
The weight of deformation at observation point in two stageswas calculated The results are shown in the Table 1
63 Results of Information Entropy of Overall DeformationRange of cloud fall of the displacement at G21 27 and28 is shown in Figure 13 The boundary values of mostdangerous information entropy from 2003 to 2013 are shownin Tables 2ndash5The absolute value means the degree of dangerdownstream is set as negative value In the significance level120572 = 005 7 kinds of common distribution (lognormal distri-bution normal distribution uniform distribution triangulardistribution exponential distribution 120574 distribution and 120573distribution) were used to carry on hypothesis test for 1198780
119898119895
1198781
119898119895 1198770119898119895 and 1198771
119898119895 with K-S method The maximum 119863
119899
of each sequencewas obtained and comparedwith the criticalvalue 119863005
119899of the significant level 005 to determine the type
Mathematical Problems in Engineering 9
Table 1 The weight table of observation point
Observation point Altitude The period of temperature rise The period of temperature dropG21 134m 0357 039227 118m 0331 031628 107m 0312 0292
Table 2 The lower limit of the most dangerous information entropy of downstream overall deformation
Year 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013Information entropy 04784 04837 04767 04784 04969 04887 04969 05102 05082 04799 04739
Table 3 The upper limit of the most dangerous information entropy of downstream overall deformation
Year 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013Information entropy 04791 04845 04770 04786 04971 04899 04974 05186 05161 04817 04769
Table 4 The lower limit of the most dangerous information entropy of upstream overall deformation
Year 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013Information entropy minus05433 minus05491 minus04773 minus05083 minus05067 minus04784 minus05080 minus04839 minus04836 minus04825 minus05020
Table 5 The lower limit of the most dangerous information entropy of upstream overall deformation
Year 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013Information entropy minus05339 minus05440 minus04770 minus05079 minus05055 minus04770 minus05040 minus04822 minus04806 minus04754 minus04766
Cloud droplets
minus15
minus14
minus13
minus12
minus11
minus1
minus09
minus08
minus07
minus06
minus05
minus04
minus03
minus02
minus01
010
02
03
04
05
06
07
08
09 1
00102030405060708091
Cer
tain
ty
G21 27 28
Figure 13 Range of cloud fall of the displacement at G21 27 and 28
of the best distribution K-S test results are shown in Table 6Multistage fuzzy warning values are presented in Table 7
K-S test shows that 1198780119898119895 1198781119898119895 1198770119898119895 and 1198771
119898119895 satisfy
normal distributionThe probability density function of the sequence is
119891 (119909) =1
radic21205871205902exp(minus
(119909 minus 120583)2
120590) (21)
Parameter values of (21) are presented in Table 7In downstream deformation if 120572 = 5 the primary
warning indicator is (04763 04775) if 120572 = 1 thesecondary warning indicator is (04757 04763) 120575lowast is used as
the boundary value when two indicators overlap In the caseof upstream deformation if 120572 = 5 the primary warningindicator is (minus04779 minus04768) if 120572 = 1 the secondarywarning indicator is (minus04768 minus04763) (Table 8)
64 The Structure Calculation of Monitoring Index of DamHorizontal Displacement According to the actual situationthree-dimensional finite element model of the dam is estab-lished According to the structure and basic geologicalconditions of 21 dam section the scope of finite elementcalculation model can be got as follows taking 15 times ashigh dam in the upstream direction taking 15 times as highdam in the upstream direction and taking 1 time as highdam below the dam foundation The model is constitutedof 11445 nodes and 8571 units The unit type is 6 sides 8nodes isoparametric element The deformation observationdata analysis shows the dam under the condition of lowtemperature and highwater level there is larger displacementin the downstream when in high temperature and in lowwater level there is larger displacement in the upstream(Table 9 Figure 14) In view of the actual situation of thedam observation and data analysis results the load conditionselection is as follows (Table 10 Figures 15 and 16)
Working condition 1 normal water level 1290m andmaximum temperature dropWorking condition 2 dead water level 1220m andmaximum temperature rise
10 Mathematical Problems in Engineering
Table 6 K-S test results
Probability distribution 1198780
1198981198951198781
1198981198951198770
1198981198951198771
119898119895
Lognormal distribution 026 011 023 028Normal distribution 011 008 026 022Uniform distribution 074 155 088 068Triangular distribution 053 054 059 061Exponential distribution 041 042 055 035120574 distribution 037 036 031 033120573 distribution 068 073 086 063119863005
119899034 034 029 029
The most reasonable probability distribution Normal distribution Normal distribution Normal distribution Normal distribution
Table 7 Parameter values of the probability density function
Data series Parameter values120583 120590
2
1198780
119898119895 0488355 00129
1198781
119898119895 0490627 00151
1198770
119898119895 minus050210 00249
1198771
119898119895 minus049674 00245
Figure 14 Finite element model of the dam
The primary warning indicators of concrete dam wereobtained by calculation methods for structures If the infor-mation entropy of the overall deformation reached 04770the dammoveddownstreamand in the state of primarywarn-ing If the information entropy of the overall deformationreached minus04773 the dam moved upstream and in the stateof primary warning (Table 11) They all fell into intervalscalculated by fuzzy methodologyThe analysis shows that themethod brought up in this paper is reasonable and scientificAlso the analysis shows the physical meaning of the fuzzywarning indexUnder the action of the unfavorable load com-bination and the influence of the complex random factorsthe maximum entropy andminimum information entropy ofthe overall deformation lie in this interval Considering theinfluences of random factorsmultistage fuzzywarning valueswere receptive and safe
In this paper the overall deformation of 21 buttresswas analyzed through the theoretical method proposedInfluences of random factors on the warning value wereconsidered and multistage fuzzy warning values were deter-mined If the information entropy of overall deformationis in (04763 04775) the dam moved downstream and in
minus2399e minus 003minus1296e minus 003minus1940e minus 0049084e minus 0042011e minus 0033113e minus 0034216e minus 0035318e minus 0036421e minus 0037523e minus 0038625e minus 003
XY
Z
Figure 15The results of finite element calculation ofworking condi-tion 1
XY
Z minus6007e minus 004minus5014e minus 004minus4021e minus 004minus3028e minus 004minus2035e minus 004minus1042e minus 004minus4872e minus 0069443e minus 0051937e minus 0042930e minus 0043923e minus 004
Figure 16 The results of finite element calculation of workingcondition 2
the state of primary warning If the information entropy ofoverall deformation is in (04757 04763) the dam moveddownstream and in the state of secondary warning If theinformation entropy of overall deformation is in (minus04779minus04768) the dam moved upstream and in the state ofprimary warning If the information entropy of the over-all deformation is in (minus04768 minus04763) the dam movedupstream and in the state of secondary warning
7 Conclusion
This paper presented multistage warning indicators of con-crete dam space and considered the influences of complex
Mathematical Problems in Engineering 11
Table 8 Multistage fuzzy warning values of concrete dam
The direction of deformation Confidence levelThe primary warning indicator The secondary warning indicator
Downstream (04763 04775) (04757 04763)Upstream (minus04779 minus04768) (minus04768 minus04763)
Table 9 Material parameter of the dam
Structure Density (kgm) Poissonrsquos ratio Elastic modulus (GPa)Concrete face slab 2400 0167 24Buttress 2400 0167 24Partition wall 2400 0167 24Reinforced concrete block 2400 0160 22Foundation rock mass 2700 0175 12Diorite-dyke 2000 03 115Crushed zone 2000 03 029Horizontal joints 2000 03 07
Table 10 The results of finite element calculation of displacementof observation point (mm)
Observation point Working condition 1 Working condition 2G21 0785 minus059927 0654 minus038628 0356 0114
Table 11 The primary warning indicators of concrete dam
The direction of deformation The primary warning indicatorDownstream 04770Upstream minus04773
random factors The results of the specific studies are asfollows
(1) Influences of fuzziness and randomness of randomfactors on the long-term service of dam were dis-cussed a fuzziness indicator that measures the influ-ence of random factors on monitoring value wasconstructed through cloud model
(2) Equivalent model of overall deformation was pro-posed In the model the overall deformation ofdam was regarded as a deformation system whereeach observation point had different contributions(weights) and affected one another Based on entropytheory a space information entropy that can measurethe overall deformation condition was established
(3) Multistage warning indicators against overall defor-mation of concrete dam under the influences of fuzzi-ness and randomness were determined and nondeter-ministic optimal control of the indicator was achievedto improve the competence of warning against thedeformation of concrete dam
Competing Interests
No conflict of interests exits in the submission of this paper
Acknowledgments
This paper was financially supported by National Natu-ral Science Foundation of China (Grants nos 4132300151139001 51579086 51379068 51279052 51579083 and51209077) Jiangsu Natural Science Foundation (Grants nosBK20140039 and BK2012036) Research Fund for the Doc-toral Program of Higher Education of China (Grant no20130094110010) the Ministry of Water Resources PublicWelfare Industry Research Special Fund Project (Grantsnos 201201038 and 201301061) Jiangsu Province ldquo333 High-Level Personnel Training Projectrdquo (Grants nos BRA2011179and BRA2011145) Project Funded by the Priority AcademicProgram Development of Jiangsu Higher Education Institu-tions (Grant no YS11001) Jiangsu Province ldquo333 High-LevelPersonnel Training Projectrdquo (Grant no 2017-B08037) JiangsuProvince ldquoSix Talent Peaksrdquo Project (Grant no JY-008) andFundamental Research Funds for the Central Universities(Grants nos 2016B04114 and 2015B25414)
References
[1] F BenedettoGGiunta andLMastroeni ldquoAmaximumentropymethod to assess the predictability of financial and commoditypricesrdquo Digital Signal Processing vol 46 pp 19ndash31 2015
[2] S Muraro A Madaschi and A Gajo ldquoOn the reliability of 3Dnumerical analyses on passive piles used for slope stabilisationin frictional soilsrdquo Geotechnique vol 64 no 6 pp 486ndash4922014
[3] W J Yoon and K S Park ldquoA study on the market instabilityindex and risk warning levels in early warning system foreconomic crisisrdquo Digital Signal Processing vol 29 pp 35ndash442014
[4] D Tonini ldquoObserved behavior of several leakier arch damsrdquoJournal of the Power Division vol 82 no 12 pp 135ndash139 1956
12 Mathematical Problems in Engineering
[5] R Salgado and D Kim ldquoReliability analysis of load andresistance factor design of slopesrdquo Journal of Geotechnical andGeoenvironmental Engineering vol 140 no 1 pp 57ndash73 2014
[6] Y C Gu and S J Wang ldquoStudy on the early-warning thresholdof structural instability unexpected accidents of reservoir damrdquoJournal of Hydraulic Engineering vol 40 no 12 pp 1467ndash14722009
[7] H Z Su F Wang and H P Liu ldquoEarly-warning index for damservice behavior based on POT modelrdquo Journal of HydraulicEngineering vol 43 no 8 pp 974ndash986 2012
[8] L Chen Mechanics performance with parameters changing inspace amp monitoring model of roller compacted concrete dam[PhD thesis] Hohai University 2006
[9] A B Li L F Zhang Q L Wang B Li Z Li and Y WangldquoInformation theory in nonlinear error growth dynamics andits application to predictability taking the Lorenz system as anexamplerdquo Science China Earth Sciences vol 56 no 8 pp 1413ndash1421 2013
[10] O Victor ldquoThe theory of cloudsrdquo Library Journal vol 132 no13 p 62 2007
[11] M Yang H Z Su and X Q Yan ldquoComputation and analysis ofhigh rocky slope safety in a water conservancy projectrdquoDiscreteDynamics in Nature and Society vol 2015 Article ID 197579 11pages 2015
[12] C Ricotta and G C Avena ldquoEvaluating the degree of fuzzi-ness of thematic maps with a generalized entropy functiona methodological outlookrdquo International Journal of RemoteSensing vol 23 no 20 pp 4519ndash4523 2002
[13] V I Khvorostyanov and J A Curry ldquoParameterization ofhomogeneous ice nucleation for cloud and climate modelsbased on classical nucleation theoryrdquo Atmospheric Chemistryand Physics vol 12 no 19 pp 9275ndash9302 2012
[14] H Z Su Z P Wen and Z R Wu ldquoStudy on an intelligentinference engine in early-warning system of damhealthrdquoWaterResources Management vol 25 no 6 pp 1545ndash1563 2011
[15] M Kaminski ldquoProbabilistic entropy in homogenization ofthe periodic fiber-reinforced composites with random elasticparametersrdquo International Journal for Numerical Methods inEngineering vol 90 no 8 pp 939ndash954 2012
[16] E Czogała and J Łeski ldquoApplication of entropy and energymeasures of fuzziness to processing of ECG signalrdquo Fuzzy Setsand Systems vol 97 no 1 pp 9ndash18 1998
[17] K Wu and J L Jin ldquoProjection pursuit model for evaluation ofregion water resource security based on changeable weight andinformationrdquo Resources and Environment in the Yangtze Basinvol 20 no 9 pp 1085ndash1091 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Mathematical Problems in Engineering
G21 G9 G4G24
25
30
2631
28
27
3233
Figure 8 The arrangement of observation point
Ups
tream
wat
er le
vel
623
198
4
121
419
89
66
1995
112
620
00
519
200
6
119
201
1
11
1979
Date
120124128132
Figure 9 The process line of upstream water level
11
2007
11
2008
11
2006
123
120
08
11
2005
123
120
09
123
120
10
123
120
11
123
020
12
123
020
13
Data
05
101520253035
Tem
pera
ture
Figure 10 The process line of temperature
62 Calculating the Contribution of Deformation at theObservation Point to the Overall Deformation Figures 9-10 show the upstream stage hydrograph and temperaturestage hydrograph respectively The water stage remainedunchanged whereas the temperature changed in the annualcycle Figure 11 shows the relationship between informationentropy of overall deformation and temperature in 2007 Anegative correlation exists between the overall deformationof 21 buttress and temperature that is an increase in tem-perature corresponds to the decrease in upstream or down-stream displacement and a decrease in temperature meansan increase in the upstream or downstream displacementFigure 12 shows the correlation between the displacementat observation point G21 and the temperature The overalldeformation law is roughly identical with that at observationpoint G21
Figure 12 reveals that the temperature obviously influ-enced overall deformation that is the temperature canchange the contribution of single observation point to theoverall deformation Temperature change was divided intothe stage of temperature rise and stage of temperature drop
2011
11
2011
21
2011
31
2011
41
2011
51
2011
61
2011
71
2011
81
2011
91
2011
10
1
2011
11
1
2011
12
1
TemperatureDeformed entropy
0070
140210280
Tem
pera
ture
minus080minus076minus072minus068minus064minus060
Def
orm
ed en
tropy
Figure 11 The relationship between information entropy of overalldeformation and temperature in 2011
2011
11
2011
13
1
2011
32
2011
41
2011
51
2011
53
1
2011
63
0
2011
73
0
2011
82
9
2011
92
8
2011
10
28
2011
11
27
2011
12
27
Horizontal displacementTemperature
05
101520253035
Tem
pera
ture
minus16minus12minus08minus04004
Hor
izon
tal
disp
lace
men
t
Figure 12 The relationship between horizontal displacement andtemperature of observation point G21 in 2011
The weight of deformation at observation point in two stageswas calculated The results are shown in the Table 1
63 Results of Information Entropy of Overall DeformationRange of cloud fall of the displacement at G21 27 and28 is shown in Figure 13 The boundary values of mostdangerous information entropy from 2003 to 2013 are shownin Tables 2ndash5The absolute value means the degree of dangerdownstream is set as negative value In the significance level120572 = 005 7 kinds of common distribution (lognormal distri-bution normal distribution uniform distribution triangulardistribution exponential distribution 120574 distribution and 120573distribution) were used to carry on hypothesis test for 1198780
119898119895
1198781
119898119895 1198770119898119895 and 1198771
119898119895 with K-S method The maximum 119863
119899
of each sequencewas obtained and comparedwith the criticalvalue 119863005
119899of the significant level 005 to determine the type
Mathematical Problems in Engineering 9
Table 1 The weight table of observation point
Observation point Altitude The period of temperature rise The period of temperature dropG21 134m 0357 039227 118m 0331 031628 107m 0312 0292
Table 2 The lower limit of the most dangerous information entropy of downstream overall deformation
Year 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013Information entropy 04784 04837 04767 04784 04969 04887 04969 05102 05082 04799 04739
Table 3 The upper limit of the most dangerous information entropy of downstream overall deformation
Year 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013Information entropy 04791 04845 04770 04786 04971 04899 04974 05186 05161 04817 04769
Table 4 The lower limit of the most dangerous information entropy of upstream overall deformation
Year 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013Information entropy minus05433 minus05491 minus04773 minus05083 minus05067 minus04784 minus05080 minus04839 minus04836 minus04825 minus05020
Table 5 The lower limit of the most dangerous information entropy of upstream overall deformation
Year 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013Information entropy minus05339 minus05440 minus04770 minus05079 minus05055 minus04770 minus05040 minus04822 minus04806 minus04754 minus04766
Cloud droplets
minus15
minus14
minus13
minus12
minus11
minus1
minus09
minus08
minus07
minus06
minus05
minus04
minus03
minus02
minus01
010
02
03
04
05
06
07
08
09 1
00102030405060708091
Cer
tain
ty
G21 27 28
Figure 13 Range of cloud fall of the displacement at G21 27 and 28
of the best distribution K-S test results are shown in Table 6Multistage fuzzy warning values are presented in Table 7
K-S test shows that 1198780119898119895 1198781119898119895 1198770119898119895 and 1198771
119898119895 satisfy
normal distributionThe probability density function of the sequence is
119891 (119909) =1
radic21205871205902exp(minus
(119909 minus 120583)2
120590) (21)
Parameter values of (21) are presented in Table 7In downstream deformation if 120572 = 5 the primary
warning indicator is (04763 04775) if 120572 = 1 thesecondary warning indicator is (04757 04763) 120575lowast is used as
the boundary value when two indicators overlap In the caseof upstream deformation if 120572 = 5 the primary warningindicator is (minus04779 minus04768) if 120572 = 1 the secondarywarning indicator is (minus04768 minus04763) (Table 8)
64 The Structure Calculation of Monitoring Index of DamHorizontal Displacement According to the actual situationthree-dimensional finite element model of the dam is estab-lished According to the structure and basic geologicalconditions of 21 dam section the scope of finite elementcalculation model can be got as follows taking 15 times ashigh dam in the upstream direction taking 15 times as highdam in the upstream direction and taking 1 time as highdam below the dam foundation The model is constitutedof 11445 nodes and 8571 units The unit type is 6 sides 8nodes isoparametric element The deformation observationdata analysis shows the dam under the condition of lowtemperature and highwater level there is larger displacementin the downstream when in high temperature and in lowwater level there is larger displacement in the upstream(Table 9 Figure 14) In view of the actual situation of thedam observation and data analysis results the load conditionselection is as follows (Table 10 Figures 15 and 16)
Working condition 1 normal water level 1290m andmaximum temperature dropWorking condition 2 dead water level 1220m andmaximum temperature rise
10 Mathematical Problems in Engineering
Table 6 K-S test results
Probability distribution 1198780
1198981198951198781
1198981198951198770
1198981198951198771
119898119895
Lognormal distribution 026 011 023 028Normal distribution 011 008 026 022Uniform distribution 074 155 088 068Triangular distribution 053 054 059 061Exponential distribution 041 042 055 035120574 distribution 037 036 031 033120573 distribution 068 073 086 063119863005
119899034 034 029 029
The most reasonable probability distribution Normal distribution Normal distribution Normal distribution Normal distribution
Table 7 Parameter values of the probability density function
Data series Parameter values120583 120590
2
1198780
119898119895 0488355 00129
1198781
119898119895 0490627 00151
1198770
119898119895 minus050210 00249
1198771
119898119895 minus049674 00245
Figure 14 Finite element model of the dam
The primary warning indicators of concrete dam wereobtained by calculation methods for structures If the infor-mation entropy of the overall deformation reached 04770the dammoveddownstreamand in the state of primarywarn-ing If the information entropy of the overall deformationreached minus04773 the dam moved upstream and in the stateof primary warning (Table 11) They all fell into intervalscalculated by fuzzy methodologyThe analysis shows that themethod brought up in this paper is reasonable and scientificAlso the analysis shows the physical meaning of the fuzzywarning indexUnder the action of the unfavorable load com-bination and the influence of the complex random factorsthe maximum entropy andminimum information entropy ofthe overall deformation lie in this interval Considering theinfluences of random factorsmultistage fuzzywarning valueswere receptive and safe
In this paper the overall deformation of 21 buttresswas analyzed through the theoretical method proposedInfluences of random factors on the warning value wereconsidered and multistage fuzzy warning values were deter-mined If the information entropy of overall deformationis in (04763 04775) the dam moved downstream and in
minus2399e minus 003minus1296e minus 003minus1940e minus 0049084e minus 0042011e minus 0033113e minus 0034216e minus 0035318e minus 0036421e minus 0037523e minus 0038625e minus 003
XY
Z
Figure 15The results of finite element calculation ofworking condi-tion 1
XY
Z minus6007e minus 004minus5014e minus 004minus4021e minus 004minus3028e minus 004minus2035e minus 004minus1042e minus 004minus4872e minus 0069443e minus 0051937e minus 0042930e minus 0043923e minus 004
Figure 16 The results of finite element calculation of workingcondition 2
the state of primary warning If the information entropy ofoverall deformation is in (04757 04763) the dam moveddownstream and in the state of secondary warning If theinformation entropy of overall deformation is in (minus04779minus04768) the dam moved upstream and in the state ofprimary warning If the information entropy of the over-all deformation is in (minus04768 minus04763) the dam movedupstream and in the state of secondary warning
7 Conclusion
This paper presented multistage warning indicators of con-crete dam space and considered the influences of complex
Mathematical Problems in Engineering 11
Table 8 Multistage fuzzy warning values of concrete dam
The direction of deformation Confidence levelThe primary warning indicator The secondary warning indicator
Downstream (04763 04775) (04757 04763)Upstream (minus04779 minus04768) (minus04768 minus04763)
Table 9 Material parameter of the dam
Structure Density (kgm) Poissonrsquos ratio Elastic modulus (GPa)Concrete face slab 2400 0167 24Buttress 2400 0167 24Partition wall 2400 0167 24Reinforced concrete block 2400 0160 22Foundation rock mass 2700 0175 12Diorite-dyke 2000 03 115Crushed zone 2000 03 029Horizontal joints 2000 03 07
Table 10 The results of finite element calculation of displacementof observation point (mm)
Observation point Working condition 1 Working condition 2G21 0785 minus059927 0654 minus038628 0356 0114
Table 11 The primary warning indicators of concrete dam
The direction of deformation The primary warning indicatorDownstream 04770Upstream minus04773
random factors The results of the specific studies are asfollows
(1) Influences of fuzziness and randomness of randomfactors on the long-term service of dam were dis-cussed a fuzziness indicator that measures the influ-ence of random factors on monitoring value wasconstructed through cloud model
(2) Equivalent model of overall deformation was pro-posed In the model the overall deformation ofdam was regarded as a deformation system whereeach observation point had different contributions(weights) and affected one another Based on entropytheory a space information entropy that can measurethe overall deformation condition was established
(3) Multistage warning indicators against overall defor-mation of concrete dam under the influences of fuzzi-ness and randomness were determined and nondeter-ministic optimal control of the indicator was achievedto improve the competence of warning against thedeformation of concrete dam
Competing Interests
No conflict of interests exits in the submission of this paper
Acknowledgments
This paper was financially supported by National Natu-ral Science Foundation of China (Grants nos 4132300151139001 51579086 51379068 51279052 51579083 and51209077) Jiangsu Natural Science Foundation (Grants nosBK20140039 and BK2012036) Research Fund for the Doc-toral Program of Higher Education of China (Grant no20130094110010) the Ministry of Water Resources PublicWelfare Industry Research Special Fund Project (Grantsnos 201201038 and 201301061) Jiangsu Province ldquo333 High-Level Personnel Training Projectrdquo (Grants nos BRA2011179and BRA2011145) Project Funded by the Priority AcademicProgram Development of Jiangsu Higher Education Institu-tions (Grant no YS11001) Jiangsu Province ldquo333 High-LevelPersonnel Training Projectrdquo (Grant no 2017-B08037) JiangsuProvince ldquoSix Talent Peaksrdquo Project (Grant no JY-008) andFundamental Research Funds for the Central Universities(Grants nos 2016B04114 and 2015B25414)
References
[1] F BenedettoGGiunta andLMastroeni ldquoAmaximumentropymethod to assess the predictability of financial and commoditypricesrdquo Digital Signal Processing vol 46 pp 19ndash31 2015
[2] S Muraro A Madaschi and A Gajo ldquoOn the reliability of 3Dnumerical analyses on passive piles used for slope stabilisationin frictional soilsrdquo Geotechnique vol 64 no 6 pp 486ndash4922014
[3] W J Yoon and K S Park ldquoA study on the market instabilityindex and risk warning levels in early warning system foreconomic crisisrdquo Digital Signal Processing vol 29 pp 35ndash442014
[4] D Tonini ldquoObserved behavior of several leakier arch damsrdquoJournal of the Power Division vol 82 no 12 pp 135ndash139 1956
12 Mathematical Problems in Engineering
[5] R Salgado and D Kim ldquoReliability analysis of load andresistance factor design of slopesrdquo Journal of Geotechnical andGeoenvironmental Engineering vol 140 no 1 pp 57ndash73 2014
[6] Y C Gu and S J Wang ldquoStudy on the early-warning thresholdof structural instability unexpected accidents of reservoir damrdquoJournal of Hydraulic Engineering vol 40 no 12 pp 1467ndash14722009
[7] H Z Su F Wang and H P Liu ldquoEarly-warning index for damservice behavior based on POT modelrdquo Journal of HydraulicEngineering vol 43 no 8 pp 974ndash986 2012
[8] L Chen Mechanics performance with parameters changing inspace amp monitoring model of roller compacted concrete dam[PhD thesis] Hohai University 2006
[9] A B Li L F Zhang Q L Wang B Li Z Li and Y WangldquoInformation theory in nonlinear error growth dynamics andits application to predictability taking the Lorenz system as anexamplerdquo Science China Earth Sciences vol 56 no 8 pp 1413ndash1421 2013
[10] O Victor ldquoThe theory of cloudsrdquo Library Journal vol 132 no13 p 62 2007
[11] M Yang H Z Su and X Q Yan ldquoComputation and analysis ofhigh rocky slope safety in a water conservancy projectrdquoDiscreteDynamics in Nature and Society vol 2015 Article ID 197579 11pages 2015
[12] C Ricotta and G C Avena ldquoEvaluating the degree of fuzzi-ness of thematic maps with a generalized entropy functiona methodological outlookrdquo International Journal of RemoteSensing vol 23 no 20 pp 4519ndash4523 2002
[13] V I Khvorostyanov and J A Curry ldquoParameterization ofhomogeneous ice nucleation for cloud and climate modelsbased on classical nucleation theoryrdquo Atmospheric Chemistryand Physics vol 12 no 19 pp 9275ndash9302 2012
[14] H Z Su Z P Wen and Z R Wu ldquoStudy on an intelligentinference engine in early-warning system of damhealthrdquoWaterResources Management vol 25 no 6 pp 1545ndash1563 2011
[15] M Kaminski ldquoProbabilistic entropy in homogenization ofthe periodic fiber-reinforced composites with random elasticparametersrdquo International Journal for Numerical Methods inEngineering vol 90 no 8 pp 939ndash954 2012
[16] E Czogała and J Łeski ldquoApplication of entropy and energymeasures of fuzziness to processing of ECG signalrdquo Fuzzy Setsand Systems vol 97 no 1 pp 9ndash18 1998
[17] K Wu and J L Jin ldquoProjection pursuit model for evaluation ofregion water resource security based on changeable weight andinformationrdquo Resources and Environment in the Yangtze Basinvol 20 no 9 pp 1085ndash1091 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 9
Table 1 The weight table of observation point
Observation point Altitude The period of temperature rise The period of temperature dropG21 134m 0357 039227 118m 0331 031628 107m 0312 0292
Table 2 The lower limit of the most dangerous information entropy of downstream overall deformation
Year 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013Information entropy 04784 04837 04767 04784 04969 04887 04969 05102 05082 04799 04739
Table 3 The upper limit of the most dangerous information entropy of downstream overall deformation
Year 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013Information entropy 04791 04845 04770 04786 04971 04899 04974 05186 05161 04817 04769
Table 4 The lower limit of the most dangerous information entropy of upstream overall deformation
Year 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013Information entropy minus05433 minus05491 minus04773 minus05083 minus05067 minus04784 minus05080 minus04839 minus04836 minus04825 minus05020
Table 5 The lower limit of the most dangerous information entropy of upstream overall deformation
Year 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013Information entropy minus05339 minus05440 minus04770 minus05079 minus05055 minus04770 minus05040 minus04822 minus04806 minus04754 minus04766
Cloud droplets
minus15
minus14
minus13
minus12
minus11
minus1
minus09
minus08
minus07
minus06
minus05
minus04
minus03
minus02
minus01
010
02
03
04
05
06
07
08
09 1
00102030405060708091
Cer
tain
ty
G21 27 28
Figure 13 Range of cloud fall of the displacement at G21 27 and 28
of the best distribution K-S test results are shown in Table 6Multistage fuzzy warning values are presented in Table 7
K-S test shows that 1198780119898119895 1198781119898119895 1198770119898119895 and 1198771
119898119895 satisfy
normal distributionThe probability density function of the sequence is
119891 (119909) =1
radic21205871205902exp(minus
(119909 minus 120583)2
120590) (21)
Parameter values of (21) are presented in Table 7In downstream deformation if 120572 = 5 the primary
warning indicator is (04763 04775) if 120572 = 1 thesecondary warning indicator is (04757 04763) 120575lowast is used as
the boundary value when two indicators overlap In the caseof upstream deformation if 120572 = 5 the primary warningindicator is (minus04779 minus04768) if 120572 = 1 the secondarywarning indicator is (minus04768 minus04763) (Table 8)
64 The Structure Calculation of Monitoring Index of DamHorizontal Displacement According to the actual situationthree-dimensional finite element model of the dam is estab-lished According to the structure and basic geologicalconditions of 21 dam section the scope of finite elementcalculation model can be got as follows taking 15 times ashigh dam in the upstream direction taking 15 times as highdam in the upstream direction and taking 1 time as highdam below the dam foundation The model is constitutedof 11445 nodes and 8571 units The unit type is 6 sides 8nodes isoparametric element The deformation observationdata analysis shows the dam under the condition of lowtemperature and highwater level there is larger displacementin the downstream when in high temperature and in lowwater level there is larger displacement in the upstream(Table 9 Figure 14) In view of the actual situation of thedam observation and data analysis results the load conditionselection is as follows (Table 10 Figures 15 and 16)
Working condition 1 normal water level 1290m andmaximum temperature dropWorking condition 2 dead water level 1220m andmaximum temperature rise
10 Mathematical Problems in Engineering
Table 6 K-S test results
Probability distribution 1198780
1198981198951198781
1198981198951198770
1198981198951198771
119898119895
Lognormal distribution 026 011 023 028Normal distribution 011 008 026 022Uniform distribution 074 155 088 068Triangular distribution 053 054 059 061Exponential distribution 041 042 055 035120574 distribution 037 036 031 033120573 distribution 068 073 086 063119863005
119899034 034 029 029
The most reasonable probability distribution Normal distribution Normal distribution Normal distribution Normal distribution
Table 7 Parameter values of the probability density function
Data series Parameter values120583 120590
2
1198780
119898119895 0488355 00129
1198781
119898119895 0490627 00151
1198770
119898119895 minus050210 00249
1198771
119898119895 minus049674 00245
Figure 14 Finite element model of the dam
The primary warning indicators of concrete dam wereobtained by calculation methods for structures If the infor-mation entropy of the overall deformation reached 04770the dammoveddownstreamand in the state of primarywarn-ing If the information entropy of the overall deformationreached minus04773 the dam moved upstream and in the stateof primary warning (Table 11) They all fell into intervalscalculated by fuzzy methodologyThe analysis shows that themethod brought up in this paper is reasonable and scientificAlso the analysis shows the physical meaning of the fuzzywarning indexUnder the action of the unfavorable load com-bination and the influence of the complex random factorsthe maximum entropy andminimum information entropy ofthe overall deformation lie in this interval Considering theinfluences of random factorsmultistage fuzzywarning valueswere receptive and safe
In this paper the overall deformation of 21 buttresswas analyzed through the theoretical method proposedInfluences of random factors on the warning value wereconsidered and multistage fuzzy warning values were deter-mined If the information entropy of overall deformationis in (04763 04775) the dam moved downstream and in
minus2399e minus 003minus1296e minus 003minus1940e minus 0049084e minus 0042011e minus 0033113e minus 0034216e minus 0035318e minus 0036421e minus 0037523e minus 0038625e minus 003
XY
Z
Figure 15The results of finite element calculation ofworking condi-tion 1
XY
Z minus6007e minus 004minus5014e minus 004minus4021e minus 004minus3028e minus 004minus2035e minus 004minus1042e minus 004minus4872e minus 0069443e minus 0051937e minus 0042930e minus 0043923e minus 004
Figure 16 The results of finite element calculation of workingcondition 2
the state of primary warning If the information entropy ofoverall deformation is in (04757 04763) the dam moveddownstream and in the state of secondary warning If theinformation entropy of overall deformation is in (minus04779minus04768) the dam moved upstream and in the state ofprimary warning If the information entropy of the over-all deformation is in (minus04768 minus04763) the dam movedupstream and in the state of secondary warning
7 Conclusion
This paper presented multistage warning indicators of con-crete dam space and considered the influences of complex
Mathematical Problems in Engineering 11
Table 8 Multistage fuzzy warning values of concrete dam
The direction of deformation Confidence levelThe primary warning indicator The secondary warning indicator
Downstream (04763 04775) (04757 04763)Upstream (minus04779 minus04768) (minus04768 minus04763)
Table 9 Material parameter of the dam
Structure Density (kgm) Poissonrsquos ratio Elastic modulus (GPa)Concrete face slab 2400 0167 24Buttress 2400 0167 24Partition wall 2400 0167 24Reinforced concrete block 2400 0160 22Foundation rock mass 2700 0175 12Diorite-dyke 2000 03 115Crushed zone 2000 03 029Horizontal joints 2000 03 07
Table 10 The results of finite element calculation of displacementof observation point (mm)
Observation point Working condition 1 Working condition 2G21 0785 minus059927 0654 minus038628 0356 0114
Table 11 The primary warning indicators of concrete dam
The direction of deformation The primary warning indicatorDownstream 04770Upstream minus04773
random factors The results of the specific studies are asfollows
(1) Influences of fuzziness and randomness of randomfactors on the long-term service of dam were dis-cussed a fuzziness indicator that measures the influ-ence of random factors on monitoring value wasconstructed through cloud model
(2) Equivalent model of overall deformation was pro-posed In the model the overall deformation ofdam was regarded as a deformation system whereeach observation point had different contributions(weights) and affected one another Based on entropytheory a space information entropy that can measurethe overall deformation condition was established
(3) Multistage warning indicators against overall defor-mation of concrete dam under the influences of fuzzi-ness and randomness were determined and nondeter-ministic optimal control of the indicator was achievedto improve the competence of warning against thedeformation of concrete dam
Competing Interests
No conflict of interests exits in the submission of this paper
Acknowledgments
This paper was financially supported by National Natu-ral Science Foundation of China (Grants nos 4132300151139001 51579086 51379068 51279052 51579083 and51209077) Jiangsu Natural Science Foundation (Grants nosBK20140039 and BK2012036) Research Fund for the Doc-toral Program of Higher Education of China (Grant no20130094110010) the Ministry of Water Resources PublicWelfare Industry Research Special Fund Project (Grantsnos 201201038 and 201301061) Jiangsu Province ldquo333 High-Level Personnel Training Projectrdquo (Grants nos BRA2011179and BRA2011145) Project Funded by the Priority AcademicProgram Development of Jiangsu Higher Education Institu-tions (Grant no YS11001) Jiangsu Province ldquo333 High-LevelPersonnel Training Projectrdquo (Grant no 2017-B08037) JiangsuProvince ldquoSix Talent Peaksrdquo Project (Grant no JY-008) andFundamental Research Funds for the Central Universities(Grants nos 2016B04114 and 2015B25414)
References
[1] F BenedettoGGiunta andLMastroeni ldquoAmaximumentropymethod to assess the predictability of financial and commoditypricesrdquo Digital Signal Processing vol 46 pp 19ndash31 2015
[2] S Muraro A Madaschi and A Gajo ldquoOn the reliability of 3Dnumerical analyses on passive piles used for slope stabilisationin frictional soilsrdquo Geotechnique vol 64 no 6 pp 486ndash4922014
[3] W J Yoon and K S Park ldquoA study on the market instabilityindex and risk warning levels in early warning system foreconomic crisisrdquo Digital Signal Processing vol 29 pp 35ndash442014
[4] D Tonini ldquoObserved behavior of several leakier arch damsrdquoJournal of the Power Division vol 82 no 12 pp 135ndash139 1956
12 Mathematical Problems in Engineering
[5] R Salgado and D Kim ldquoReliability analysis of load andresistance factor design of slopesrdquo Journal of Geotechnical andGeoenvironmental Engineering vol 140 no 1 pp 57ndash73 2014
[6] Y C Gu and S J Wang ldquoStudy on the early-warning thresholdof structural instability unexpected accidents of reservoir damrdquoJournal of Hydraulic Engineering vol 40 no 12 pp 1467ndash14722009
[7] H Z Su F Wang and H P Liu ldquoEarly-warning index for damservice behavior based on POT modelrdquo Journal of HydraulicEngineering vol 43 no 8 pp 974ndash986 2012
[8] L Chen Mechanics performance with parameters changing inspace amp monitoring model of roller compacted concrete dam[PhD thesis] Hohai University 2006
[9] A B Li L F Zhang Q L Wang B Li Z Li and Y WangldquoInformation theory in nonlinear error growth dynamics andits application to predictability taking the Lorenz system as anexamplerdquo Science China Earth Sciences vol 56 no 8 pp 1413ndash1421 2013
[10] O Victor ldquoThe theory of cloudsrdquo Library Journal vol 132 no13 p 62 2007
[11] M Yang H Z Su and X Q Yan ldquoComputation and analysis ofhigh rocky slope safety in a water conservancy projectrdquoDiscreteDynamics in Nature and Society vol 2015 Article ID 197579 11pages 2015
[12] C Ricotta and G C Avena ldquoEvaluating the degree of fuzzi-ness of thematic maps with a generalized entropy functiona methodological outlookrdquo International Journal of RemoteSensing vol 23 no 20 pp 4519ndash4523 2002
[13] V I Khvorostyanov and J A Curry ldquoParameterization ofhomogeneous ice nucleation for cloud and climate modelsbased on classical nucleation theoryrdquo Atmospheric Chemistryand Physics vol 12 no 19 pp 9275ndash9302 2012
[14] H Z Su Z P Wen and Z R Wu ldquoStudy on an intelligentinference engine in early-warning system of damhealthrdquoWaterResources Management vol 25 no 6 pp 1545ndash1563 2011
[15] M Kaminski ldquoProbabilistic entropy in homogenization ofthe periodic fiber-reinforced composites with random elasticparametersrdquo International Journal for Numerical Methods inEngineering vol 90 no 8 pp 939ndash954 2012
[16] E Czogała and J Łeski ldquoApplication of entropy and energymeasures of fuzziness to processing of ECG signalrdquo Fuzzy Setsand Systems vol 97 no 1 pp 9ndash18 1998
[17] K Wu and J L Jin ldquoProjection pursuit model for evaluation ofregion water resource security based on changeable weight andinformationrdquo Resources and Environment in the Yangtze Basinvol 20 no 9 pp 1085ndash1091 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Mathematical Problems in Engineering
Table 6 K-S test results
Probability distribution 1198780
1198981198951198781
1198981198951198770
1198981198951198771
119898119895
Lognormal distribution 026 011 023 028Normal distribution 011 008 026 022Uniform distribution 074 155 088 068Triangular distribution 053 054 059 061Exponential distribution 041 042 055 035120574 distribution 037 036 031 033120573 distribution 068 073 086 063119863005
119899034 034 029 029
The most reasonable probability distribution Normal distribution Normal distribution Normal distribution Normal distribution
Table 7 Parameter values of the probability density function
Data series Parameter values120583 120590
2
1198780
119898119895 0488355 00129
1198781
119898119895 0490627 00151
1198770
119898119895 minus050210 00249
1198771
119898119895 minus049674 00245
Figure 14 Finite element model of the dam
The primary warning indicators of concrete dam wereobtained by calculation methods for structures If the infor-mation entropy of the overall deformation reached 04770the dammoveddownstreamand in the state of primarywarn-ing If the information entropy of the overall deformationreached minus04773 the dam moved upstream and in the stateof primary warning (Table 11) They all fell into intervalscalculated by fuzzy methodologyThe analysis shows that themethod brought up in this paper is reasonable and scientificAlso the analysis shows the physical meaning of the fuzzywarning indexUnder the action of the unfavorable load com-bination and the influence of the complex random factorsthe maximum entropy andminimum information entropy ofthe overall deformation lie in this interval Considering theinfluences of random factorsmultistage fuzzywarning valueswere receptive and safe
In this paper the overall deformation of 21 buttresswas analyzed through the theoretical method proposedInfluences of random factors on the warning value wereconsidered and multistage fuzzy warning values were deter-mined If the information entropy of overall deformationis in (04763 04775) the dam moved downstream and in
minus2399e minus 003minus1296e minus 003minus1940e minus 0049084e minus 0042011e minus 0033113e minus 0034216e minus 0035318e minus 0036421e minus 0037523e minus 0038625e minus 003
XY
Z
Figure 15The results of finite element calculation ofworking condi-tion 1
XY
Z minus6007e minus 004minus5014e minus 004minus4021e minus 004minus3028e minus 004minus2035e minus 004minus1042e minus 004minus4872e minus 0069443e minus 0051937e minus 0042930e minus 0043923e minus 004
Figure 16 The results of finite element calculation of workingcondition 2
the state of primary warning If the information entropy ofoverall deformation is in (04757 04763) the dam moveddownstream and in the state of secondary warning If theinformation entropy of overall deformation is in (minus04779minus04768) the dam moved upstream and in the state ofprimary warning If the information entropy of the over-all deformation is in (minus04768 minus04763) the dam movedupstream and in the state of secondary warning
7 Conclusion
This paper presented multistage warning indicators of con-crete dam space and considered the influences of complex
Mathematical Problems in Engineering 11
Table 8 Multistage fuzzy warning values of concrete dam
The direction of deformation Confidence levelThe primary warning indicator The secondary warning indicator
Downstream (04763 04775) (04757 04763)Upstream (minus04779 minus04768) (minus04768 minus04763)
Table 9 Material parameter of the dam
Structure Density (kgm) Poissonrsquos ratio Elastic modulus (GPa)Concrete face slab 2400 0167 24Buttress 2400 0167 24Partition wall 2400 0167 24Reinforced concrete block 2400 0160 22Foundation rock mass 2700 0175 12Diorite-dyke 2000 03 115Crushed zone 2000 03 029Horizontal joints 2000 03 07
Table 10 The results of finite element calculation of displacementof observation point (mm)
Observation point Working condition 1 Working condition 2G21 0785 minus059927 0654 minus038628 0356 0114
Table 11 The primary warning indicators of concrete dam
The direction of deformation The primary warning indicatorDownstream 04770Upstream minus04773
random factors The results of the specific studies are asfollows
(1) Influences of fuzziness and randomness of randomfactors on the long-term service of dam were dis-cussed a fuzziness indicator that measures the influ-ence of random factors on monitoring value wasconstructed through cloud model
(2) Equivalent model of overall deformation was pro-posed In the model the overall deformation ofdam was regarded as a deformation system whereeach observation point had different contributions(weights) and affected one another Based on entropytheory a space information entropy that can measurethe overall deformation condition was established
(3) Multistage warning indicators against overall defor-mation of concrete dam under the influences of fuzzi-ness and randomness were determined and nondeter-ministic optimal control of the indicator was achievedto improve the competence of warning against thedeformation of concrete dam
Competing Interests
No conflict of interests exits in the submission of this paper
Acknowledgments
This paper was financially supported by National Natu-ral Science Foundation of China (Grants nos 4132300151139001 51579086 51379068 51279052 51579083 and51209077) Jiangsu Natural Science Foundation (Grants nosBK20140039 and BK2012036) Research Fund for the Doc-toral Program of Higher Education of China (Grant no20130094110010) the Ministry of Water Resources PublicWelfare Industry Research Special Fund Project (Grantsnos 201201038 and 201301061) Jiangsu Province ldquo333 High-Level Personnel Training Projectrdquo (Grants nos BRA2011179and BRA2011145) Project Funded by the Priority AcademicProgram Development of Jiangsu Higher Education Institu-tions (Grant no YS11001) Jiangsu Province ldquo333 High-LevelPersonnel Training Projectrdquo (Grant no 2017-B08037) JiangsuProvince ldquoSix Talent Peaksrdquo Project (Grant no JY-008) andFundamental Research Funds for the Central Universities(Grants nos 2016B04114 and 2015B25414)
References
[1] F BenedettoGGiunta andLMastroeni ldquoAmaximumentropymethod to assess the predictability of financial and commoditypricesrdquo Digital Signal Processing vol 46 pp 19ndash31 2015
[2] S Muraro A Madaschi and A Gajo ldquoOn the reliability of 3Dnumerical analyses on passive piles used for slope stabilisationin frictional soilsrdquo Geotechnique vol 64 no 6 pp 486ndash4922014
[3] W J Yoon and K S Park ldquoA study on the market instabilityindex and risk warning levels in early warning system foreconomic crisisrdquo Digital Signal Processing vol 29 pp 35ndash442014
[4] D Tonini ldquoObserved behavior of several leakier arch damsrdquoJournal of the Power Division vol 82 no 12 pp 135ndash139 1956
12 Mathematical Problems in Engineering
[5] R Salgado and D Kim ldquoReliability analysis of load andresistance factor design of slopesrdquo Journal of Geotechnical andGeoenvironmental Engineering vol 140 no 1 pp 57ndash73 2014
[6] Y C Gu and S J Wang ldquoStudy on the early-warning thresholdof structural instability unexpected accidents of reservoir damrdquoJournal of Hydraulic Engineering vol 40 no 12 pp 1467ndash14722009
[7] H Z Su F Wang and H P Liu ldquoEarly-warning index for damservice behavior based on POT modelrdquo Journal of HydraulicEngineering vol 43 no 8 pp 974ndash986 2012
[8] L Chen Mechanics performance with parameters changing inspace amp monitoring model of roller compacted concrete dam[PhD thesis] Hohai University 2006
[9] A B Li L F Zhang Q L Wang B Li Z Li and Y WangldquoInformation theory in nonlinear error growth dynamics andits application to predictability taking the Lorenz system as anexamplerdquo Science China Earth Sciences vol 56 no 8 pp 1413ndash1421 2013
[10] O Victor ldquoThe theory of cloudsrdquo Library Journal vol 132 no13 p 62 2007
[11] M Yang H Z Su and X Q Yan ldquoComputation and analysis ofhigh rocky slope safety in a water conservancy projectrdquoDiscreteDynamics in Nature and Society vol 2015 Article ID 197579 11pages 2015
[12] C Ricotta and G C Avena ldquoEvaluating the degree of fuzzi-ness of thematic maps with a generalized entropy functiona methodological outlookrdquo International Journal of RemoteSensing vol 23 no 20 pp 4519ndash4523 2002
[13] V I Khvorostyanov and J A Curry ldquoParameterization ofhomogeneous ice nucleation for cloud and climate modelsbased on classical nucleation theoryrdquo Atmospheric Chemistryand Physics vol 12 no 19 pp 9275ndash9302 2012
[14] H Z Su Z P Wen and Z R Wu ldquoStudy on an intelligentinference engine in early-warning system of damhealthrdquoWaterResources Management vol 25 no 6 pp 1545ndash1563 2011
[15] M Kaminski ldquoProbabilistic entropy in homogenization ofthe periodic fiber-reinforced composites with random elasticparametersrdquo International Journal for Numerical Methods inEngineering vol 90 no 8 pp 939ndash954 2012
[16] E Czogała and J Łeski ldquoApplication of entropy and energymeasures of fuzziness to processing of ECG signalrdquo Fuzzy Setsand Systems vol 97 no 1 pp 9ndash18 1998
[17] K Wu and J L Jin ldquoProjection pursuit model for evaluation ofregion water resource security based on changeable weight andinformationrdquo Resources and Environment in the Yangtze Basinvol 20 no 9 pp 1085ndash1091 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 11
Table 8 Multistage fuzzy warning values of concrete dam
The direction of deformation Confidence levelThe primary warning indicator The secondary warning indicator
Downstream (04763 04775) (04757 04763)Upstream (minus04779 minus04768) (minus04768 minus04763)
Table 9 Material parameter of the dam
Structure Density (kgm) Poissonrsquos ratio Elastic modulus (GPa)Concrete face slab 2400 0167 24Buttress 2400 0167 24Partition wall 2400 0167 24Reinforced concrete block 2400 0160 22Foundation rock mass 2700 0175 12Diorite-dyke 2000 03 115Crushed zone 2000 03 029Horizontal joints 2000 03 07
Table 10 The results of finite element calculation of displacementof observation point (mm)
Observation point Working condition 1 Working condition 2G21 0785 minus059927 0654 minus038628 0356 0114
Table 11 The primary warning indicators of concrete dam
The direction of deformation The primary warning indicatorDownstream 04770Upstream minus04773
random factors The results of the specific studies are asfollows
(1) Influences of fuzziness and randomness of randomfactors on the long-term service of dam were dis-cussed a fuzziness indicator that measures the influ-ence of random factors on monitoring value wasconstructed through cloud model
(2) Equivalent model of overall deformation was pro-posed In the model the overall deformation ofdam was regarded as a deformation system whereeach observation point had different contributions(weights) and affected one another Based on entropytheory a space information entropy that can measurethe overall deformation condition was established
(3) Multistage warning indicators against overall defor-mation of concrete dam under the influences of fuzzi-ness and randomness were determined and nondeter-ministic optimal control of the indicator was achievedto improve the competence of warning against thedeformation of concrete dam
Competing Interests
No conflict of interests exits in the submission of this paper
Acknowledgments
This paper was financially supported by National Natu-ral Science Foundation of China (Grants nos 4132300151139001 51579086 51379068 51279052 51579083 and51209077) Jiangsu Natural Science Foundation (Grants nosBK20140039 and BK2012036) Research Fund for the Doc-toral Program of Higher Education of China (Grant no20130094110010) the Ministry of Water Resources PublicWelfare Industry Research Special Fund Project (Grantsnos 201201038 and 201301061) Jiangsu Province ldquo333 High-Level Personnel Training Projectrdquo (Grants nos BRA2011179and BRA2011145) Project Funded by the Priority AcademicProgram Development of Jiangsu Higher Education Institu-tions (Grant no YS11001) Jiangsu Province ldquo333 High-LevelPersonnel Training Projectrdquo (Grant no 2017-B08037) JiangsuProvince ldquoSix Talent Peaksrdquo Project (Grant no JY-008) andFundamental Research Funds for the Central Universities(Grants nos 2016B04114 and 2015B25414)
References
[1] F BenedettoGGiunta andLMastroeni ldquoAmaximumentropymethod to assess the predictability of financial and commoditypricesrdquo Digital Signal Processing vol 46 pp 19ndash31 2015
[2] S Muraro A Madaschi and A Gajo ldquoOn the reliability of 3Dnumerical analyses on passive piles used for slope stabilisationin frictional soilsrdquo Geotechnique vol 64 no 6 pp 486ndash4922014
[3] W J Yoon and K S Park ldquoA study on the market instabilityindex and risk warning levels in early warning system foreconomic crisisrdquo Digital Signal Processing vol 29 pp 35ndash442014
[4] D Tonini ldquoObserved behavior of several leakier arch damsrdquoJournal of the Power Division vol 82 no 12 pp 135ndash139 1956
12 Mathematical Problems in Engineering
[5] R Salgado and D Kim ldquoReliability analysis of load andresistance factor design of slopesrdquo Journal of Geotechnical andGeoenvironmental Engineering vol 140 no 1 pp 57ndash73 2014
[6] Y C Gu and S J Wang ldquoStudy on the early-warning thresholdof structural instability unexpected accidents of reservoir damrdquoJournal of Hydraulic Engineering vol 40 no 12 pp 1467ndash14722009
[7] H Z Su F Wang and H P Liu ldquoEarly-warning index for damservice behavior based on POT modelrdquo Journal of HydraulicEngineering vol 43 no 8 pp 974ndash986 2012
[8] L Chen Mechanics performance with parameters changing inspace amp monitoring model of roller compacted concrete dam[PhD thesis] Hohai University 2006
[9] A B Li L F Zhang Q L Wang B Li Z Li and Y WangldquoInformation theory in nonlinear error growth dynamics andits application to predictability taking the Lorenz system as anexamplerdquo Science China Earth Sciences vol 56 no 8 pp 1413ndash1421 2013
[10] O Victor ldquoThe theory of cloudsrdquo Library Journal vol 132 no13 p 62 2007
[11] M Yang H Z Su and X Q Yan ldquoComputation and analysis ofhigh rocky slope safety in a water conservancy projectrdquoDiscreteDynamics in Nature and Society vol 2015 Article ID 197579 11pages 2015
[12] C Ricotta and G C Avena ldquoEvaluating the degree of fuzzi-ness of thematic maps with a generalized entropy functiona methodological outlookrdquo International Journal of RemoteSensing vol 23 no 20 pp 4519ndash4523 2002
[13] V I Khvorostyanov and J A Curry ldquoParameterization ofhomogeneous ice nucleation for cloud and climate modelsbased on classical nucleation theoryrdquo Atmospheric Chemistryand Physics vol 12 no 19 pp 9275ndash9302 2012
[14] H Z Su Z P Wen and Z R Wu ldquoStudy on an intelligentinference engine in early-warning system of damhealthrdquoWaterResources Management vol 25 no 6 pp 1545ndash1563 2011
[15] M Kaminski ldquoProbabilistic entropy in homogenization ofthe periodic fiber-reinforced composites with random elasticparametersrdquo International Journal for Numerical Methods inEngineering vol 90 no 8 pp 939ndash954 2012
[16] E Czogała and J Łeski ldquoApplication of entropy and energymeasures of fuzziness to processing of ECG signalrdquo Fuzzy Setsand Systems vol 97 no 1 pp 9ndash18 1998
[17] K Wu and J L Jin ldquoProjection pursuit model for evaluation ofregion water resource security based on changeable weight andinformationrdquo Resources and Environment in the Yangtze Basinvol 20 no 9 pp 1085ndash1091 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
12 Mathematical Problems in Engineering
[5] R Salgado and D Kim ldquoReliability analysis of load andresistance factor design of slopesrdquo Journal of Geotechnical andGeoenvironmental Engineering vol 140 no 1 pp 57ndash73 2014
[6] Y C Gu and S J Wang ldquoStudy on the early-warning thresholdof structural instability unexpected accidents of reservoir damrdquoJournal of Hydraulic Engineering vol 40 no 12 pp 1467ndash14722009
[7] H Z Su F Wang and H P Liu ldquoEarly-warning index for damservice behavior based on POT modelrdquo Journal of HydraulicEngineering vol 43 no 8 pp 974ndash986 2012
[8] L Chen Mechanics performance with parameters changing inspace amp monitoring model of roller compacted concrete dam[PhD thesis] Hohai University 2006
[9] A B Li L F Zhang Q L Wang B Li Z Li and Y WangldquoInformation theory in nonlinear error growth dynamics andits application to predictability taking the Lorenz system as anexamplerdquo Science China Earth Sciences vol 56 no 8 pp 1413ndash1421 2013
[10] O Victor ldquoThe theory of cloudsrdquo Library Journal vol 132 no13 p 62 2007
[11] M Yang H Z Su and X Q Yan ldquoComputation and analysis ofhigh rocky slope safety in a water conservancy projectrdquoDiscreteDynamics in Nature and Society vol 2015 Article ID 197579 11pages 2015
[12] C Ricotta and G C Avena ldquoEvaluating the degree of fuzzi-ness of thematic maps with a generalized entropy functiona methodological outlookrdquo International Journal of RemoteSensing vol 23 no 20 pp 4519ndash4523 2002
[13] V I Khvorostyanov and J A Curry ldquoParameterization ofhomogeneous ice nucleation for cloud and climate modelsbased on classical nucleation theoryrdquo Atmospheric Chemistryand Physics vol 12 no 19 pp 9275ndash9302 2012
[14] H Z Su Z P Wen and Z R Wu ldquoStudy on an intelligentinference engine in early-warning system of damhealthrdquoWaterResources Management vol 25 no 6 pp 1545ndash1563 2011
[15] M Kaminski ldquoProbabilistic entropy in homogenization ofthe periodic fiber-reinforced composites with random elasticparametersrdquo International Journal for Numerical Methods inEngineering vol 90 no 8 pp 939ndash954 2012
[16] E Czogała and J Łeski ldquoApplication of entropy and energymeasures of fuzziness to processing of ECG signalrdquo Fuzzy Setsand Systems vol 97 no 1 pp 9ndash18 1998
[17] K Wu and J L Jin ldquoProjection pursuit model for evaluation ofregion water resource security based on changeable weight andinformationrdquo Resources and Environment in the Yangtze Basinvol 20 no 9 pp 1085ndash1091 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of